Significant Digits for Wednesday, Nov. 27, 2019

You’re reading Significant Digits, a daily digest of the numbers tucked inside the news. Today’s number is 800-288-8372 — the legendary Butterball Turkey Talk-Line, which answered 11,265 calls last year on Thanksgiving Day.


33,000 excess deaths

A new report published in the Journal of the American Medical Association says more Americans are dying at younger ages despite greater spending on health care in the U.S. than in any other country. Mortality and life expectancy rates for Americans between the ages of 25 and 64 are getting worse, with increases in suicides, drug overdoses and alcohol-related liver disease, among other factors. The study estimated that there were 33,000 excess midlife deaths between 2010 and 2017, and the negative trend was found across gender, race and ethnicity. [The Washington Post]


40 pizzas

The founder of Papa John’s isn’t impressed with the pizza chain since his ouster 18 months ago. John Schnatter told Louisville, Kentucky, TV station WDRB that he had eaten 40 pizzas in the past 30 days, but “it’s not the same pizza.” Schnatter said that new company chief Rob Lynch has “no pizza experience,” that two of the company’s board members should be put in jail, and that “the day of reckoning will come.” [WDRB]


20 million unique customers

If you’re thinking of checking out online luxury stores this week to get going on your holiday shopping, there’s a good chance you’ll encounter technology from a software customer service company called Powerfront. Its flagship product helps sales representatives see, chat with and track online customers using cartoon-like avatars. Powerfront says the service has become incredibly popular among hundreds of clients and is capable of processing huge amounts of information: approximately 20 million unique customers every 24 hours. [The New York Times]


143 Taser incidents

Bad behavior in school usually means a trip to the principal’s office, not a hit from a Taser. A new investigation from HuffPost found at least 143 incidents since 2011 in which children as young as 11 were Tasered by school officers, even though Tasers can be deadly. Some of the incidents were for the protection of children, but there are also cases on the other end of the disciplinary spectrum, like an instance last year when a deputy used a Taser to wake a sleeping student. [HuffPost]


3 times as much mercury

Sometimes even the way pollution spreads has a way of surprising scientists. A new study from the University of California, Santa Cruz found three times as much mercury in mountain lions in the Santa Cruz Mountains as in the same animals inland. The source of the mercury is likely coal-burning plants; the metal lands in the ocean and sinks to the bottom, then bacteria convert it into a fat-soluble form that is picked up by the coastal fog and carried into the mountains. The mercury is absorbed by lichen, which are consumed by deer, which are eaten by mountain lions. [San Francisco Chronicle]


40 percent of household income

Men report higher levels of psychological distress when they are the sole household earners — distress that decreases as their wives earn more money. But that comes with a catch, according to a report published last month in the Personality and Social Psychology Bulletin. Economist Dr. Joanna Syrda found that the husbands’ stress hit a low point when their wives earned 40 percent of the household income but began to increase again as the wives began to outearn them. One silver living: If women made more money than men when they got married, the men didn’t have the same concerns. The long-term study looked at thousands of couples in the U.S. over 14 years. [Good Morning America]


SigDigs: Nov. 27, 2019

What’s Going To Happen With Impeachment? 2020? It’s Another Game Of Buy, Sell, Hold!

Welcome to FiveThirtyEight’s weekly politics chat. The transcript below has been lightly edited.


sarahf (Sarah Frostenson, politics editor): It certainly has been quite the news cycle, what with five impeachment hearings last week and a Democratic debate, so in light of the Thanksgiving holiday, this week we’re taking a step back to play a good old-fashioned game of buy/sell/hold with PredictIt prop bets (plus some I made up).

We’ll talk where impeachment is headed and our best guesses for what’s happening in the Democratic primary. PredictIt’s prices (given in cents) as of noon Eastern on Tuesday were translated into probabilities. (We know that’s not exactly right, but it’s close enough.)

And in case you forgot how to play buy/sell/hold

  • Buy means “I think the chances of this happening are higher than indicated.”
  • Sell means “I think they’re lower.”
  • Hold means “I’m a coward and am unwilling to take a stand.”

OK, to kick us off. Buy, sell or hold: President Trump will be impeached in his first term. (73 percent).

nrakich (Nathaniel Rakich, elections analyst): The chances of that happening are much higher than 73 percent. Probably like 95 percent. So I guess I’m buying? (This game always confuses me.)

perry (Perry Bacon Jr., senior writer): I would hold. I still think there is some chance Democrats back down — since the polling suggests support for impeachment remains divided along party lines. (So if you’re a Democrat in a district Trump won, being pro-impeachment is probably a controversial view among your constituents.) The House Judiciary Committee is having a hearing exploring the constitutional grounds for impeachment on Dec. 4, though, so it does suggest that Democrats haven’t gone too wobbly on the impeachment plan.

nrakich: Interesting, Perry!

Wouldn’t it look terrible for Democrats to back down at this stage? The polls haven’t changed much since the Ukraine scandal broke — but they’re much better for impeachment than they were before it broke. And impeachment is still more popular than it is unpopular.

geoffrey.skelley (Geoffrey Skelley, elections analyst): I think I’ll buy at that price, given that Democrats need 216 of their 233 members to vote yes (considering vacancies), but as Perry says, there are some Democrats who’ve sounded somewhat less certain recently.

For instance, Michigan Democratic Rep. Brenda Lawrence came out in favor of censure instead of impeachment, although she walked that back on Tuesday, saying she still supports impeachment.

nrakich: Hmm. I grant you that there are some Democratic members who might want to vote “no” on impeachment out of self-interest. But the self-interest for the Democratic Party as a whole is obviously not to say, “Our bad, the president we’ve been railing against as a criminal for three years actually isn’t a criminal after all.”

sarahf: We still need some more polls before we understand the effect of the hearings, but as FiveThirtyEight’s editor-in-chief Nate Silver pointed out, support overall is slowly ticking back upward. Now, that might not last, or it might just be noise, but as he says, it pushes back against this idea that what Democrats have done is wildly unpopular.

So I’d buy, too.

But OK. Buy, sell or hold: The Senate will convict Trump on impeachment in his first term. (13 percent)

perry: Sell. There is just no chance of a removal vote. None. Maybe Trump resigns if somehow impeachment support gets really high, but I don’t see Republicans voting for his removal. I don’t think there is any real chance Trump resigns either.

nrakich: I think the chances that the Senate removes Trump are not literally zero percent, but they aren’t as high as 13 percent. (I’d maybe say 5 percent?) So yeah, sell.

geoffrey.skelley: I would sell at 13 percent. It’s just such a high bar to get 20 Republicans to vote to remove Trump, which would be needed for removal, assuming all Democrats vote in favor. But as FiveThirtyEight contributor Lee Drutman wrote, if the GOP starts opposing Trump, it might be hard to predict in advance. It could happen fast and all at once.

sarahf: Right, and there’s not really any signs of congressional Republicans breaking with Trump at this point, right?

Or in other words, is there anything that would make you not sell at this point?

geoffrey.skelley: I mean, GOP Rep. Will Hurd was one of just 13 Republicans in the House to vote to oppose Trump’s national emergency declaration over the U.S.-Mexico border, but he’s on the Intelligence Committee and looks like a “no” on impeachment. If he’s not breaking, I’m not sure who will.

But if someone like Hurd was more open to impeachment, especially because he’s not running for reelection, I’d take that as a sign that a break with Trump might be possible.

nrakich: But even so, Hurd isn’t your average congressional Republican, and I don’t think there’s any sign of those average Republicans breaking with Trump, which is what would need to happen for removal.

geoffrey.skelley: I don’t have a probability, but buy, sell, hold: Two GOP senators vote to remove Trump from office.

perry: Sell. There could be one senator (Trump skeptic Mitt Romney of Utah) who votes for removal. But I think the more likely number is zero Republicans back removal.

nrakich: I agree. It’s more likely that a Democrat (i.e., Sen. Joe Manchin of West Virginia) votes not to remove than that a Republican votes to remove.

sarahf: 🔥

geoffrey.skelley: Oh, I like that take.

nrakich: Removing a president is just such a drastic step. Even if Romney or moderate Sen. Susan Collins disagree with the president, that doesn’t mean they would vote to oust him from office.

perry: I could definitely see Manchin and Sen. Doug Jones of Alabama voting against removal.

nrakich: Jones would be in a tricky place — he is actually not that moderate. But you’d have to think a vote to remove would be political suicide in Alabama.

perry: Exactly.

geoffrey.skelley: Given he’s trying to win reelection in a state Trump will probably carry by 20+ points, voting to remove might make his defeat a done deal.

perry: On the other hand, Jones may know he’s very likely to lose his reelection bid in 2020 anyway, so maybe he votes for removal because that’s his real view. He might assume he has a better chance of being selected as attorney general for a Democratic president in 2021 than winning another term in the Senate from Alabama. (Jones was the U.S. attorney for the Northern District of Alabama from 1997-2001.)

geoffrey.skelley: Maybe a yes vote by Jones is a sign he knows he’s a goner. A no vote signals he’s still in it to win it.

sarahf: But 56 percent of Americans said in our poll with Ipsos that Trump had committed an impeachable offense, and there were some Republicans, too, open to the idea of impeachment. You don’t think if public opinion continues to tick upward, that wouldn’t convince any Republican senator to break?

perry: But in our impeachment tracker, support for impeachment is at 12 percent among Republicans. Why would any Republican in Congress vote for impeachment?

geoffrey.skelley: To Sarah’s point, based on our poll with Ipsos, it looks like about 28 percent of Republicans might be open to impeachment if they come to believe Trump withheld aid or tried to cover it up, compared to the 11 to 12 percent of Republicans supporting impeachment in our tracker. But that still would mean 70 to 75 percent oppose impeachment, and a bunch of GOP senators don’t want to risk Republican voters’ wrath in primaries in 2020 (or in the future).

sarahf: OK, let’s leave impeachment alone for now — looks like there will be plenty more to speculate on soon enough with the first House Judiciary Committee hearing scheduled for Dec. 4 — and move on to Democratic primary wagers.

South Bend, Indiana, Mayor Pete Buttigieg has been on a bit of an upswing lately. Strong performance in the last debate, some good polls in Iowa and New Hampshire, and the betting markets currently have him winning the Iowa caucuses (37 percent). Buy, sell or hold?

nrakich: Ooh, that’s a tough one. I guess I’d hold — I just don’t know what will happen, so I don’t have strong evidence to argue one way or another.

perry: Sell. I think Buttigieg is doing very well in Iowa. Perhaps he is the favorite there. But I would say he has a 30 percent chance to win the state, Warren a 25 percent chance, Biden and Sanders both a 20 percent shot and maybe a 5 percent chance for someone who is not one of those four.

geoffrey.skelley: I’m going to sell only because it seems like there are four candidates truly in the mix in Iowa, and I’m not sure anyone’s chances are above 30 percent out of the maybe 95 percent of outcomes that involve one of them winning.

Perry and I are very much on the same page!

sarahf: That’s interesting, Perry, regarding the order you’ve put the candidates in — PredictIt has it has Buttigieg, Sanders, Warren and then Biden in terms of their odds winning Iowa. So a bit more bullish for Sanders than I think the conventional wisdom holds, but right about where you’d expect Biden given his lackluster poll numbers in the state.

So tell me, since you and Geoff seem to be of the same mind, why do you think Sanders stands a chance of winning Iowa?

geoffrey.skelley: I think it mainly comes down to the fact that the vote could be sufficiently fragmented, and that means Sanders could win with 22 to 23 percent or something.

perry: I’m not really bullish on Sanders. I just think Buttigieg’s chances of winning Iowa are less than 37 percent.

geoffrey.skelley: The Democratic caucuses are also difficult to project because if a voter’s candidate isn’t clearing a certain threshold at their caucus site — usually 15 percent — that means a voter has to support another candidate who has cleared it. This is called the viability threshold, and it means that voters’ second-choice picks will be really important in Iowa, which is one reason Sanders could do so well. He is the backup pick for 13 percent of likely caucus goers, according to a CNN/Des Moines Register/Mediacom Iowa poll earlier this month.

sarahf: On that note, I’m going to go off book and give you all a wager from my ~mind~ (dangerous, I know). Let’s talk about how many candidates win 15 percent or more of the statewide vote in Iowa, which is the mechanism by which candidates win delegates (this is different than the viability threshold Geoff mentioned above, which looks at each caucus site individually — confusing, I know). Since 1992 when the 15 percent threshold was first introduced, there’s never been a primary or caucus (in Iowa or elsewhere) where four or more candidates have earned more than 15 percent of the statewide vote, but there have been many instances where three candidates have cleared it. So, first up — buy, sell, hold: Three candidates win 15 percent of the vote in Iowa this year (45 percent).

nrakich: Yes! I would buy that, Sarah. I think three candidates will win 15 percent or more. In fact, I think there’s a pretty decent chance that four will, which, as you note, would be unprecedented.

perry: I would buy three candidates getting to 15 percent in Iowa, and I think I would buy four candidates getting to 15 percent as well.

sarahf: What would you put those odds at, Nathaniel, for four candidates clearing it?

nrakich: Oh man, totally out of thin air … I’d say there’s like a 75 percent chance that at least three candidates exceed 15 percent, and a 30 percent chance that four candidates do.

sarahf: I don’t know. It’s hard for me to know whether to buy. Every four years as political journalists we go through this whole rigamarole where we salivate over the idea of a brokered convention.

So what makes you all think four candidates is a real possibility this year compared to say, 2016 on the GOP side, when things were also crowded?

nrakich: I mean, who knows if it would lead to a brokered convention. For one thing, a step like this is probably necessary but not sufficient for convention chaos.

But I think 2020 is different from 2016 because the polling shows that four candidates are already at or above 15 percent in Iowa right now — and that’s before the reallocation of all the lower-tier candidates’ supporters.

geoffrey.skelley: I’d argue, though, that we might see more movement before the caucuses that could lead one of the four leading candidates to slump and fall short of 15 percent, leaving us with only three candidates. Right now, that looks like Biden, but maybe it’s actually Warren or one of the others.

sarahf: OK, in the vein of candidates clearing thresholds statewide to win delegates … currently, Michael Bloomberg sits at 12 percent for winning the Democratic nomination, but I’m more interested in wagers on whether he wins a state (any state) on Super Tuesday. Again from my brain — bidders beware. Buy, sell, hold (5 percent).

perry: Buy. I think Bloomberg has the chance to do well. So. Much. Money.

nrakich: But, Perry, trying to buy political office doesn’t always work if voters simply aren’t interested in what you’re selling. That said, I think 5 percent is pretty low, so I’ll buy too.

I just think things are so unpredictable after the first four states.

Maybe Biden wins all four and the primary is effectively over by Super Tuesday, in which case I’d expect Bloomberg to get no traction. But maybe Buttigieg, Warren, Biden and Sanders each wins one early state and it’s a free-for-all!

geoffrey.skelley: I remain very skeptical of Bloomberg’s chances, though I see the case people are making. It’s possible that things are such a mess after the first four states that Bloomberg could swoop in and present himself as the solution to the chaos. But I think it’s far more likely that Bloomberg’s inability to appeal to Democrats keeps him from taking advantage even if that does happen. So I’m going to sell at 5 percent that he wins any Super Tuesday state.

sarahf: It’s just such an odd strategy to me that he’s decided to skip out on the first two early states to make up ground in the other states. You’ve looked at the historical numbers, Geoff — has this ever worked?

geoffrey.skelley: To be fair, there haven’t been that many times when candidates have openly minimized campaigning in the early states — or even skipped Iowa or New Hampshire outright. But it didn’t work for Henry “Scoop” Jackson in 1976, Al Gore in 1988 or Rudy Giuliani in 2008.

sarahf: I rest my case.

nrakich: I also would be more convinced of this if Bloomberg had said he’s going to go all out to win, say, Virginia.

But spreading his efforts across multiple Super Tuesday states might make it more likely that he comes in first place in none of them.

Now, you need to win more than just one Super Tuesday state to win the nomination, so I get why he’s not doing that. But come on, Bloomberg, why won’t you change your campaign strategy to go along with our prop bet???

sarahf: Haha OK, let’s end on where things stand overall in the 2020 primary. Elizabeth Warren has long been a favorite of the betting markets, but Joe Biden is actually in the lead at the moment with a 24 percent shot. So buy, sell or hold: Biden wins the 2020 Democratic nomination?

nrakich: At 24 percent? Buy.

Biden, to me, remains the single most likely Democratic nominee — especially so now that some of the wind has been taken out of Warren’s sails.

It’s still competitive, but to me that’s good enough for Biden to have a 30-40 percent chance.

perry Hold. I’m just confused. He leads nationally but is fairly weak in Iowa and New Hampshire. And I wonder if losing in those states will affect his numbers in subsequent primaries. I just don’t know.

nrakich: My pushback to that, Perry, is that he’s only weak in the first two early states, and there “weak” just means “not in first place.” It would be entirely unsurprising to me if he gains a few points and wins both Iowa and New Hampshire in the end.

geoffrey.skelley: Based on my analysis of early primary polls earlier this year, a candidate polling at about 25 percent nationally in the second half of the year before the primary has around a 25 percent chance of winning the nomination.

I haven’t calculated things yet for the second half of this year — the year isn’t over, after all — but Biden is probably polling somewhere around 25 percent. So I think I would hold at 24 percent based on that historical record.

sarahf: Whoever defined the ground rule of holding as “I’m a coward and am unwilling to take a stand” was shortsighted. Hold confidently, Geoff!

Daniel Jones Hasn’t Figured Out Zone Coverage Yet

In Week 7 against the Arizona Cardinals, New York Giants quarterback Daniel Jones dropped back to pass on third and 13 from his own 35-yard line. Faced with a four-man rush and seven defenders in coverage, Jones waited patiently — perhaps too patiently — for his receivers to complete their routes past the first-down marker. Feeling pressure on his left, Jones stepped up in the pocket and fired the ball downfield — right into the arms of Cardinals linebacker Jordan Hicks. Hicks was playing zone coverage in the hook/curl area of the field, and he was in the perfect position to step in front of intended receiver Golden Tate.

While it might be tempting to dismiss the mistakes of a rookie QB as growing pains, the interception wasn’t an isolated case of a young quarterback making a questionable decision. It turns out that zone coverage has been a problem for Jones since he took over the Giants starting job in Week 3. Six of Jones’s eight interceptions on the year have come against zone, and he’s averaging just 5.6 yards per attempt against the coverage — worst in the league. Jones has completed passes 1.7 percentage points under what we would expect of a league-average QB against zone, according to NFL Next Gen Stats, and his QBR is an anemic 27.5. More surprising, given generational talent Saquon Barkley at running back, is Jones’s QBR of 24.9 on play-action passes against zone, which places him 30th out of 32 qualifying quarterbacks.

Jones’s poor performance on play-action is a little strange because we might expect that teams would play more Cover 1 (a form of man coverage with a single high safety) against the Giants with Barkley in the backfield, a defensive formation that allows them to bring a safety down from deep coverage to help out against the run. Through Week 12, however, that hasn’t been the case. New York opponents are playing 50.4 percent man coverage vs. 49.1 percent zone, which is almost exactly league average. The team would likely benefit from opponents playing more man coverage to match up against Barkley, as Jones is completing passes 1.9 percentage points over expected against man, with a more respectable QBR of 71.5.

Interestingly, Jones’s coverage splits are the inverse of league trends. At nearly every depth of target from zero to 30 yards, quarterbacks’ completion percentages are higher against zone coverage than against man in 2019.

The gap between zone and man is particularly pronounced on throws of 10 air yards or less. This makes intuitive sense: Defenses playing zone are typically happy to allow the short completion and rally for a tackle. But the success of man versus zone on deeper passes is more of a surprise. Zone seeks to take away the deep ball in favor of short, manageable gains. But in 2019, passers are completing a higher percentage of attempts against zone of up to at least 28 yards.

Meanwhile, play-action across the league has been quite successful against man coverage, which might come as a surprise to some NFL coaches.

Man is still the more effective coverage, but the gap between play-action and other passes is wider against man coverage than zone, making it a preferred tactic on intermediate throws.

Yet these leaguewide base rates don’t fit Jones’s statistics. Jones is an enigma — bad at things that most QBs excel at, like completing passes on play-action, yet good at some aspects of the game that many quarterbacks find extremely challenging. Against disguised coverages, for instance — coverages that start out looking like man or zone but then switch mid-play — Jones has the highest completion percentage over expected in the league4 at 10.8 percent. It’s pure zone coverage that’s his kryptonite.

Jones’s struggles against zone coverage likely explain at least some of the Giants’ disappointing year. He’ll need to show progress before next season if New York has any hope of competing with the Dallas Cowboys and Philadelphia Eagles in the NFC East. But there are some signs that he may be improving: Jones’s lone TD against zone coverage came in last week’s Week 12 loss to Chicago. A strong finish to the season might be enough for the Giants to find a reason for optimism heading into 2020.

Check out our latest NFL predictions.

Our Poll Shows That Buttigieg’s Post-Debate Bump Is Still Just His Base

The past few weeks have been pretty good for South Bend, Indiana, Mayor Pete Buttigieg: His polls are on the upswing in Iowa, he’s getting more media coverage, and he even led in a New Hampshire poll. It seems that he carried this momentum through the fifth Democratic debate as well. According to our FiveThirtyEight/Ipsos poll conducted using Ipsos’s KnowledgePanel, he saw the largest increase in the number of people considering supporting him, going from 26 percent before the debate to 32 percent afterward.

But where did that 6-point increase come from?

To answer this, we looked at a few metrics in our poll, which surveyed the same group of respondents before and after the November debate. One of the indicators we considered is how support shifted among respondents who prioritized the top five issues in our poll, which were health care, the economy and jobs, wealth and income inequality, climate change, and discrimination. As you can see in the table below, Buttigieg gained potential supporters among voters who prioritized all of the top 5 issue areas — he is the only candidate for which this was true. His gains were not marginal either, mostly around 5 percentage points.

Buttigieg gained among voters who prioritize …

Change in the share of respondents considering supporting each candidate before and after the fifth Democratic debate by which issue respondents said was most important to them, per a FiveThirtyEight/Ipsos poll

candidate Health care The economy and jobs Wealth and income inequality Climate change Racism, sexism, discrimination
Biden +1.9 +0.8 +2.4 -2.6 +3.0
Sanders +1.3 -2.9 +3.1 +0.5 +0.6
Warren +0.5 -2.3 +0.3 +2.5 -5.7
Buttigieg +5.0 +4.8 +5.7 +6.3 +3.7
Harris +5.3 +2.2 +2.2 +6.2 -3.0
Yang +1.7 +0.0 +0.8 +1.3 -0.2
Booker +3.7 +1.8 +1.6 +3.5 -0.4
Klobuchar +0.8 +1.3 +1.2 +3.1 -1.1
Gabbard +0.1 +0.6 +0.7 +0.4 +0.5
Steyer +3.8 +1.3 +0.7 +1.2 -0.3

Showing the top five issues out of a set of twelve respondents could choose from. Uses respondents’ pre-debate answer for which issue is most important. From a survey of 3,786 likely Democratic primary voters who were surveyed between Nov. 14 and Nov. 18. The same people were surveyed again from Nov. 20 to Nov. 21; 2,077 responded to the second wave.

When respondents were asked to rate candidates’ chances of beating President Trump, Buttigieg gained ground there as well, earning a post-debate average of 49.5 percent, 3 points higher than his pre-debate average. He still trails former Vice President Joe Biden (67.5 percent after the debate), Sen. Bernie Sanders (58.5 percent) and Sen. Elizabeth Warren (57.8 percent), but notably, he was the only one of these four candidates who gained in this metric — the other three either lost ground or saw no change.

But if Buttigieg was hoping his high debate marks would help him diversify his base of support, that hasn’t happened yet. The demographic cross-tabs in our poll show that he mainly made inroads among groups where he already enjoyed a disproportionate amount of support, like the college-educated, white voters and older voters. He had little success winning people over among groups where he has tended to struggle, like with black and Hispanic voters.

Buttigieg’s gains were mostly confined to his base

Share of respondents considering supporting Buttigieg in a FiveThirtyEight/Ipsos poll, broken down by demographics

Age pre-debate post-debate change
65+ 36.6% 44.5% +7.9
50-64 30.1 37.8 +7.7
18-34 19.7 24.4 +4.7
35-49 22.0 24.5 +2.4
education pre-debate post-debate change
College or higher 34.2% 41.6% +7.4
Some college 23.9 28.4 +4.5
High school or less 17.9 22.0 +4.1
race pre-debate post-debate change
White 34.5% 42.4% +7.8
Black 12.2 15.9 +3.7
Hispanic 17.4 16.3 -1.1

Only groups with a sample size of 200 or more were included. From a survey of 3,786 likely Democratic primary voters who were surveyed between Nov. 14 and Nov. 18. The same people were surveyed again from Nov. 20 to Nov. 21; 2,077 responded to the second wave.

This lack of diverse support may be a part of why Buttigieg is struggling to gain traction outside Iowa and New Hampshire and continues to sit at about 8 percent nationally, far behind the other three front-runners in the race. And if Buttigieg can’t appeal to people outside his existing base, he might have a hard time getting his numbers up any higher.

Politics Podcast: A Very FiveThirtyEight Thanksgiving

After five days of testimony, the House’s public impeachment hearings are over for now. In this installment of the FiveThirtyEight Politics podcast, the team takes stock of what we learned and the effect of the hearings on public opinion.

Also, the crew digs into the data to see if there was a conclusive winner of last week’s Democratic debate (spoiler alert: There was).

Finally, the team plays a Thanksgiving-themed game in which they try to guess what the candidates in the Democratic primary are thankful for.

You can listen to the episode by clicking the “play” button in the audio player above or by downloading it in iTunes, the ESPN App or your favorite podcast platform. If you are new to podcasts, learn how to listen.

The FiveThirtyEight Politics podcast publishes Mondays and Thursdays. Help new listeners discover the show by leaving us a rating and review on iTunes. Have a comment, question or suggestion for “good polling vs. bad polling”? Get in touch by email, on Twitter or in the comments.

FiveThirtyEight Politics Podcast: Has public opinion shifted on impeachment?

Is Aaron Rodgers Slipping? And WTF Is Happening With Carson Wentz?

sara.ziegler (Sara Ziegler, sports editor): Week 12 brought us a bunch of high-leverage games between playoff contenders: Green Bay at San Francisco, Dallas at New England and Seattle at Philadelphia. (And there’s another big game tonight between the Baltimore Ravens and the Los Angeles Rams.) So let’s talk through what we saw in those matchups and what changed in the playoff picture as a result.

Let’s start with San Francisco’s huge win over Green Bay. I’m a Niners fan this year — and I obviously root for a different team in the NFC North than the Packers — but I definitely did not see that one coming.

neil (Neil Paine, senior sportswriter): It’s getting harder and harder to doubt the Niners’ Super Bowl chances …

Salfino (Michael Salfino, FiveThirtyEight contributor): I did see it coming not because of intuition or any special insight but simply because of net yards per pass play gained minus net yards allowed. The Niners were huge there heading into Week 12 and continued putting up big numbers this week, when Aaron Rodgers generated 66 net passing yards on 38 pass plays (including sacks).

joshua.hermsmeyer (Josh Hermsmeyer, NFL analyst): It was a very impressive win by the Niners. What struck me was the job they did on Rodgers. On play-action, the most efficient play type in football, Rodgers dialed up 13 plays — good for fourth in the league so far in Week 12 — and averaged just 4 yards per dropback on those plays.

sara.ziegler: The Niners defense just looks so good.

Salfino: This was one of those games where halfway through the first quarter you knew it was over. The Packers wanted no part of the heat the 49ers were bringing.

neil: The Super Bowl will be Niners-Pats, won’t it? And the final score will be 13-3 or something.

Salfino: We often talk about explosive offensive teams, but the 49ers with that rush are an explosive defensive team.

I see the 49ers as a complete team. Great offensive mind and solid skill players on offense across the board. I know a lot of people are still skeptical of their QB, but Jimmy Garoppolo is at least average. And they just destroy their opponents’ passing games.

joshua.hermsmeyer: Weirdly, that was Rodgers’s worst game of his career by passing yardage when he’s had at least 30 pass attempts.

sara.ziegler: Wow

The Niners haven’t seemed scary offensively to me, so the 37 points they put up on Sunday took me by surprise.

They rank only 14th in total passing yards this season.

Salfino: But they’ve got George Kittle (if he’s healthy) and Deebo Samuel looking like the new Anquan Boldin plus a healthy Emmanuel Sanders. With that stable of backs plus the Kyle Shanahan system … I like their offense a lot.

joshua.hermsmeyer: They really got a lift from Emmanuel Sanders coming back. He’s a difference maker for Jimmy G.

And yes, Kittle of course.

neil: Jimmy G is almost up to average in our Elo QB ratings!

It’s taken him all season.

Salfino: I actually think Samuel is their most dangerous wide receiver now with that yards after catch number.

My defense of Jimmy G. is that he’s no worse than Matt Ryan, and Shanahan crushed it with Ryan. Plus QB wins are a real thing, I believe. He has to get some credit for his record, especially in 2017.

neil: I still think he’s a slightly-above-average QB whose defense has been killing it.

🔥

joshua.hermsmeyer: I cannot let the QB Winz comment pass! I agree that QBs are the most important piece to winning, but that stat is just terrible.

Salfino: You win a game usually by outpassing your opponent measured by net yards per attempt, and there are only two ways to do that — with your offense/QB or with your defense. So how can the QB not be the most important player to winning?

joshua.hermsmeyer: That doesn’t make QB Winz a stat!

Salfino: I’m saying that to be a good QB, you need to have a good yards per attempt or net yards per attempt, and if you do, your teams will win. So potato/tomato (as my daughter jokes).

neil: Jimmy G, for what it’s worth, is 11th in adjusted net yards per attempt, despite the Niners being second only to New England in the Simple Rating System.

To me, that indicates that the rest of the team has carried more of the load.

Salfino: I agree that the Niners’ pass defense is mostly responsible for winning this year. But I specifically mentioned 2017, and I can’t make the argument he’s much worse now.

joshua.hermsmeyer: Even last night, there were times when Jimmy couldn’t come off his first read fast enough and looked like a statue. He’s in a good scheme with the best possible support.

Salfino: I agree that he holds the ball too long. But we forget the guy has barely started any games even though it feels like he’s been around forever.

sara.ziegler: And that’s the whole problem with him! We don’t have enough of a sample size.

San Francisco has two tricky road games up next: at Baltimore this Sunday and at New Orleans the following week. How do the Niners match up against those opponents?

neil: Yes, San Francisco’s schedule gets much tougher. They played the second easiest schedule so far in terms of average opponent pregame Elo (adjusted for location); only Buffalo faced easier opponents. The Niners’ rest-of-season schedule is the toughest in the league by Elo.

Which NFL teams have the toughest remaining schedules?

Elo ranking of the hardest strength of schedule for 2019 NFL teams, for both completed and upcoming games

Elo SOS Rank Elo SOS Rank
Team record Completed upcoming Team record Completed upcoming
SF 10-1 31st 1st DEN 3-8 1st 17th
BUF 8-3 32nd 2nd CIN 0-11 18th 18th
TEN 6-5 26th 3rd MIN 8-3 21st 19th
CHI 5-6 30th 4th DET 3-7-1 9th 20th
LAR 6-4 14th 5th CAR 5-6 8th 21st
BAL 8-2 23rd 6th KC 7-4 2nd 22nd
ARI 3-7-1 15th 7th DAL 6-5 27th 23rd
NO 9-2 22nd 8th TB 4-7 7th 24th
PIT 6-5 24th 9th NYJ 4-7 25th 25th
SEA 9-2 19th 10th MIA 2-9 13th 26th
ATL 3-8 3rd 11th JAX 4-7 12th 27th
IND 6-5 11th 12th GB 8-3 4th 28th
LAC 4-7 28th 13th NE 10-1 29th 29th
OAK 6-5 16th 14th NYG 2-9 17th 30th
HOU 7-4 6th 15th CLE 5-6 10th 31st
WSH 2-9 20th 16th PHI 5-6 5th 32nd

Rankings are based on the average quality of opponent, adjusted for quarterback and location.

Salfino: No one matches up well with Baltimore. They are a unicorn team. Or if someone does match up, how would we even know until we see it? But the Niners don’t even need that game, IMO. They need to take care of business with the Saints, though. But the Saints don’t strike me as a legit Super Bowl team. They were crushed at home against the Falcons and then almost lost to Kyle Allen. And we all wrote Allen’s obit last week.

neil: I will admit I was surprised the Saints let Carolina hang around in that one.

Salfino: We have to give them a little slack with no Marcus Lattimore, I guess.

sara.ziegler: So what of the Niners’ opponent Sunday night? I don’t know what to think of Green Bay at all.

neil: Is Rodgers still ELITE??????

He hasn’t looked that way for a few games now.

joshua.hermsmeyer: The Packers have been exposed twice, and Rodgers is firmly out of the MVP conversation. Matt LeFleur said they were outcoached and outplayed. I’d be very worried.

sara.ziegler: Don’t look now, but the NFC North is basically a tossup.

neil: Heyyyyy look at that.

joshua.hermsmeyer: The Vikings are better. Sorry, Sara.

Salfino: You mean that you don’t want to look now, Sara. 🙂 I’ve been saying the Vikings are winning that division for weeks.

Kirk Cousins for MVP!

(half joking)

sara.ziegler: OMG

You need to be fully joking.

Salfino: Cousins does have numbers, Sara.

neil: Joking aside, that Packers-Vikings game in Week 16 could be huge.

And I love that matchup on Monday Night Football. So much nostalgia.

Salfino: This Packers team going into Minnesota, where that defense generally plays great, and beating the Vikings with the season on the line? No way. I don’t see it.

sara.ziegler: The Packers get the New York Giants, Washington and Chicago before the showdown with Minnesota, while the Vikings have Seattle, Detroit and the Los Angeles Chargers. So the edge has to be with the Packers to win the division, right?

Salfino: Danny Dimes, Sara.

sara.ziegler: 🙄🙄🙄

Salfino: Generational talent Saquon Barkley is averaging almost 4 yards per carry this season!

joshua.hermsmeyer: Oh yeah, generational talent, because you only get a 4.0 rushing average from a first-round pick once every 10 to 14 years, LOL.

neil: Detroit and San Diego are done.

(I don’t care that the Chargers are in L.A.)

sara.ziegler: LOL, San Diego is extremely done.

neil:

Salfino: Which Rivers child is that?

neil: LOLOLOL

sara.ziegler: Oh, wow.

neil: Gosh dang it, Salfie!

sara.ziegler: Sticking with the AFC, New England outlasted Dallas on Sunday in a game that was, for me, more about the Cowboys underperforming (and making terrible decisions at key moments) than the Patriots dominating. I just can’t get excited about this New England team, even if it’s 10-1. Which means the Patriots are certainly going to win the Super Bowl.

Salfino: The Patriots are flawed offensively but work like a hive on defense, passing off coverage from linebackers to defensive ends seamlessly, like on the third down in the red zone in the fourth quarter against the Cowboys.

neil: I know the weather was bad on Sunday, but Tom Brady has looked mediocre for a little while now.

joshua.hermsmeyer: The Pats are also one of the most man-coverage-heavy teams in the league. They used man against the Cowboys to great effect: 24 snaps in man coverage, with Dallas mustering just a 29 percent success rate and -0.42 expected points added per play. Meanwhile, they played 10 snaps of zone, and Dallas ate it up, with a 50 percent success rate and 0.26 EPA per play.

neil: This is the lowest Brady’s QB Elo rating has been (relative to average) since after the On To Cincinnati game in 2014.

And that year, they <<checks notes>> won the Super Bowl. Oh.

Salfino: The Patriots have had less than 350 yards from scrimmage in five straight games, which is their longest streak since 2003.

sara.ziegler: And yet … they keep winning!

(Except when they’re playing Baltimore.)

neil: This defense, particularly against the pass, remains incredible.

Lamar Jackson is still the only QB to play above average against their defense this season.

sara.ziegler: Jackson has a way of making good defenses look average — Seattle knows a little something about that. The Seahawks got a convincing win over the Eagles on Sunday. Seattle is in command of the first wild-card spot now, but does it have a chance for the division against the Niners?

joshua.hermsmeyer: Our model gives San Francisco a 58 percent chance to win the division and gives Seattle a 40 percent shot, so definitely a chance!

Salfino: Aren’t the Seahawks setting a record of sorts in the divergence between point differential and win-loss record?

Are they just super lucky?

neil: Yeah, Mike — the pythagorean expectation says they should only be 6-5 right now, not 9-2.

Salfino: Ross Tucker said on Twitter that Jadeveon Clowney could be out for the year? No idea if that’s true, but if it is, that’s really bad for Seattle.

joshua.hermsmeyer: If Pete Carroll is a tactical genius, it seems like his strategy is coaxing his opponents to play suboptimally, running a lot, and then relying on his elite QB to bail his team out on third and long while the other team has to punt more often in those situations.

neil: That’s the most concise, accurate description of the Seahawks I’ve ever heard.

sara.ziegler: What about the Eagles? Their goose is cooked in the wild-card race, but they still have a 37 percent chance to win the NFC East.

Salfino: Have the Eagles been beaten by flea flickers now two straight weeks? Has to be a record.

For all of Dallas’s problems, I just don’t see them losing this division to the Eagles. And Dallas’s remaining schedule isn’t exactly daunting.

sara.ziegler: Wait, Michael, you really believe in the Cowboys? Even with the play-calling choices in the game against New England?

Salfino: Well, the numbers are there for the Cowboys in the things that I think are most predictive of future wins: play success, passing success and net yards per attempt. But they have not beaten a quality team, which is very troubling. And we have to assume that the head coach is a big minus.

So I don’t know how to get my head around this team. If you believe in just the numbers, they should easily win the weak NFC East. But if you believe in coaching and intangibles, you have to expect them to struggle to beat anyone.

sara.ziegler: This was telling, I think:

Salfino: I think it’s fair to say that Jason Garrett hurt his team’s chances by not treating the third and 7 on that deep fourth-quarter drive as two-down territory. So it was not just the kick on fourth down that killed them but also the call on third down, where maybe they run if they have a four-down mentality. You can’t go from needing a touchdown to win to … needing a touchdown to win.

How Week 16′s Eagles-Cowboys game swings the playoff odds

Playoff odds if …
Team Eagles win Cowboys win
Eagles 66.4% 4.1%
Cowboys 36.0 96.4

Odds are conditional only on Week 16′s game.

neil: Here’s a fun set of contingencies for that Week 16 Eagles-Cowboys game — playoff odds according to Elo:

joshua.hermsmeyer: Wow. That would be a good time for the Cowboys to win their first big game.

neil: It’s basically a win-and-you’re-in for Dallas and do-or-die for Philly. But Philly isn’t quite in (and Dallas isn’t quite out) even if the Eagles do win.

Philly can’t feel good after these back-to-back home losses, especially with Carson Wentz playing so poorly.

joshua.hermsmeyer: The Eagles lack weapons in the passing game with all the injuries, but Wentz hasn’t exactly elevated the play of those around him. I’m still not really sure what he is. Is he a top-10 QB?

neil: Elo ranks him 17th — one spot ahead of … Jimmy G. LOL

joshua.hermsmeyer: LOL

Salfino: Wentz does not seem wired to take the positive play. He’s too much of a gunslinger when he’s on the most slow-footed offensive team I can remember. So how can he expect plays to be open downfield?

Great QBs — like we once thought Wentz was — can carry an average at best unit or at least make a poor one average. He’s not lifting the offense at all this year. He’s even holding it back.

neil: Only four QBs have been worse by yards per attempt than Wentz this year:

Ryan Fitzpatrick, Danny Dimes, Mason Rudolph and Mitchell Trubisky.

joshua.hermsmeyer: Whew

sara.ziegler: Ooof

Salfino: I just think the Eagles are mentally beaten already. The fans are booing the QB. The defense has given up 17 points in two straight weeks to the Patriots and Seahawks, and the team wasn’t really even in either game.

neil: I will say this, and we talk about it every week, but they have played a string of really tough games — six straight with an above-average Elo, going back to early November. They ought to look better against Miami, Washington and the Giants in the next few weeks. (Then again, they ought to have looked better against Seattle at home.)

joshua.hermsmeyer: Oh no, Carson:

neil: That seems right.

sara.ziegler: Yikes.

Salfino: The problem with this stat is that if guys are not open, you can’t throw a pass “on target” unless you want to risk a pick.

joshua.hermsmeyer: I agree, it doesn’t tell the whole story. But still.

Salfino: But Wentz missed an uncovered Miles Sanders on an easy flare near the goal line, and he threw the ball like crap most of the day.

neil: That was one of the worst overthrows on an easy pass I’ve ever seen, Mike.

Salfino: I do think Wentz’s problem is not taking shorter throws for positive yardage when they are there and, like Hank Stram used to say, matriculating the ball down the field. This is not a big-play offense without DeSean Jackson, and we know how vital Jackson could be to their passing success.

joshua.hermsmeyer: Dwayne Haskins had a couple pretty hideous misses yesterday as well. Then he was too busy taking selfies with fans to come out for the victory formation.

But I’m still rooting for him.

neil: Haskins’s new theme:

joshua.hermsmeyer: ded

Check out our latest NFL predictions.

How Do You Find Good NFL Defenders? By Measuring What’s Not There.

The New England Patriots’ Stephon Gilmore is widely considered one of the best cornerbacks in the NFL. Through Week 11, he had given up a 47.1 passer rating when targeted this year — to put that into perspective, a 39.6 passer rating is awarded to a quarterback who spikes the ball on every play. (Through Week 11, no qualifying starting QB in 2019 had a passer rating lower than 70.) Because of his prowess, offensive coordinators and QBs don’t even look his way: Of the 326 passing attempts against the Patriots this year, only 64 (19.6 percent) came against Gilmore.

When it comes to measuring this aspect of defensive performance, we’re only beginning to scratch the surface. That’s because individual defense is inherently difficult to assess: A player’s defensive impact may be more significant in the absence of activity, but we can only count things that do happen. To measure a defensive player, we’re always chasing ghosts, trying to count things that don’t.

But thanks to the NFL’s Next Gen Stats tracking data — and the impressive work of researchers such as Baltimore Ravens personnel analyst Sarah Mallepalle2 — we can visualize that absence of activity.

A useful example involves where on the field opponents choose to attack a defense. Every week, coaches huddle around game film, analyzing defensive schemes and players, planning how to exploit what they see. The results of that game-planning show up in the tracking data. Here are charts that show where every team’s pass defense was the most and least vulnerable in the first eight weeks of the season:3

The charts shown are scaled from blue to white and white to red, with red representing the locations where offenses attempted the most passes against the defenses, blue representing the fewest pass attempts and white representing the average number of passes. In theory, the blue areas are places where the absence of activity reveals defensive strength.

In practice, it’s more complicated to solve the problem than simply plotting the distribution of all passing attempts or completions. Some offenses pass in unique ways because of their schemes and the skill sets of their quarterbacks and receivers. Take the Seattle Seahawks, for example:

Seattle QB Russell Wilson’s pass attempts are skewed to the left.4 Why? This is likely because of the inability of D.K. Metcalf, the Seahawks’ rookie wide receiver and 2019 second-round pick, to run a full tree of routes. During the season’s first two weeks, he was targeted only while lining up on the left side of the field. Through the first half of the season, Metcalf had been targeted 45 times, and only 10 were on the right side of the field.

Given that the Seattle passing attack is skewed, we don’t want to penalize any defense’s right side5 just because those defenders had to match up against Metcalf. So to normalize for an opposing offense’s usual tendencies, I compared an offense’s pass distribution against the defense in question with that offense’s distributions against the other defenses it has faced during the 2019 season. We can consider the difference to be the defense’s relative effect on a typical offense’s gameplan.

Let’s take a deeper look at the Buffalo Bills.

The Bills defense has been outstanding, giving up 304.1 total yards and 197.8 passing yards per game through Week 11 — third best in the league in both categories. But we can see a clear disparity in how opposing offenses are attacking starting cornerbacks Tre’Davious White and Levi Wallace. White’s reputation and production seem to have discouraged offenses from throwing in his direction. So far this season, White has given up a 58.9 passer rating on 66 targets, compared with a 101.2 passer rating on 79 targets for Wallace.

The beauty of the Bills’ defense is that their star cornerbacks don’t “shadow” or follow specific opposing WRs wherever they line up. Instead, they mainly stay on one side of the field, no matter who lines up there. Through Week 8, White had lined up on the left side of the field for 97 percent of his snaps, and Wallace lined up on the right side of the field for 96 percent of his snaps.

Our chart reflects that, as well as which cornerback scares off opposing QBs and where on the field that is. Opposing offenses target the left side of the field significantly more against Buffalo (to challenge Wallace, the weaker corner) than when they play against other defenses.

This approach to analyzing defense assumes that teams react rationally to defensive weaknesses they see on tape. That assumption may not always hold true, but taking note of changing tendencies is still one of the best ways to look for those hidden absences of activity that are key to identifying good individual defense.

We still don’t know if these splits have any predictive power. But this is the next logical step in understanding passing: analyzing the horizontal level. At a minimum, these new visualizations provide an interesting new insight into how offenses change in order to attack defenses.

Check out our latest NFL predictions.

Can You Decode The Riddler Lottery?

Welcome to The Riddler. Every week, I offer up problems related to the things we hold dear around here: math, logic and probability. Two puzzles are presented each week: the Riddler Express for those of you who want something bite-size and the Riddler Classic for those of you in the slow-puzzle movement. Submit a correct answer for either,<a class="espn-footnote-link" data-footnote-id="7" href="#fn-7" data-footnote-content="<p>Important small print: Please wait until Monday to publicly share your answers. In order to 👏win 👏, I need to receive your correct answer before 11:59 p.m. Eastern time on Monday. Have a great weekend!</p> “>7 and you may get a shoutout in next week’s column. Or in this case, two weeks — Happy Thanksgiving! If you need a hint or have a favorite puzzle collecting dust in your attic, find me on Twitter.

Riddler Express

From Dave Moran comes a question about baseball’s unusual 2019 World Series:

In the World Series, one team hosts Games 1, 2, 6 and 7, while the other team hosts Games 3, 4 and 5. When the Nationals beat the Astros last month, it marked the first time in World Series history that the home team lost all seven games. On average, the home team actually wins about 54 percent of the time in baseball. Running the numbers, you’ll quickly see that seven home losses is a once-in-a-lifetime event.

But putting seven aside for a moment, what’s the probability that the home team will lose at least six consecutive games?

Extra credit: What’s the probability the home team will lose at least five consecutive games? Four consecutive games?

Submit your answer

Riddler Classic

From Charlie Cordova comes a puzzle that brings logic and number theory to the lottery:

Five friends with a lot in common are playing the Riddler Lottery, in which each must choose exactly five numbers from 1 to 70. After they all picked their numbers, the first friend notices that no number was selected by two or more friends. Unimpressed, the second friend observes that all 25 selected numbers are composite (i.e., not prime). Not to be outdone, the third friend points out that each selected number has at least two distinct prime factors. After some more thinking, the fourth friend excitedly remarks that the product of selected numbers on each ticket is exactly the same. At this point, the fifth friend is left speechless. (You can tell why all these people are friends.)

What is the product of the selected numbers on each ticket?

Extra credit: How many different ways could the friends have selected five numbers each so that all their statements are true?

Submit your answer

Solution to last week’s Riddler Express

Congratulations to 👏 Robert Reid 👏 of North Bend, Washington, winner of last week’s Riddler Express.

Last week’s puzzle was a geometric conundrum inspired by Catriona Shearer:

Inscribed semicircles

In the picture above, you were told that the lighter region (inside the larger semicircle but outside the smaller one) had an area of 7. What was the area of the darker region?

Many readers pointed out that the riddle should have been more specific — in particular, the straight edges of the two semicircles are parallel. It’s possible to inscribe a non-parallel semicircle, as shown in the animation below, but that’s beyond the scope of this question.

Different ways to inscribe one semicircle inside another

Solver Ria Skies made quick work of this puzzle with some pen and paper:

solution to semicircle riddle

Using the Pythagorean theorem, you can show that the radius of the larger semicircle is √2 times longer than the radius of the smaller semicircle. That means the area of the larger semicircle is twice that of the smaller semicircle — in other words, the area of the smaller semicircle is half the area of the larger semicircle.

Since the smaller semicircle takes up half the larger one, the remaining lighter area takes up the other half. And so the two regions — the lighter gray and the darker gray — have the same area! They both take up half the area of the larger semicircle. If the lighter region has area 7, then the darker region also has area 7.

It’s almost as if the answer was hiding in plain sight all along…

Solution to last week’s Riddler Classic

Congratulations to 👏 Jesse Cohn 👏 of St. Louis, Missouri, winner of last week’s Riddler Classic.

You were given a fair, unweighted 10-sided die with sides labeled 0 to 9 and a sheet of paper to record your score. To start the game, you rolled the die. Your current “score” was the number shown, divided by 10. For example, if you rolled a 7, then your score would be 0.7. Then, you kept rolling the die over and over again. Each time you rolled, if the digit shown by the die was less than or equal to the last digit of your score, then that roll became the new last digit of your score. Otherwise, you just went ahead and rolled again. The game ended when you rolled a zero. (If your first roll was a zero, your score was simply zero.) What was your average final score in this game?

First off, thank you to all the readers who shared links and photos of your favorite 10-sided dice and simulators of 10-sided dice. (It’s fair to say that the die in this problem likely came from a Dungeons & Dragons set.)

Getting back to the puzzle, solver Guy D. Moore made a key observation: Whenever you roll a number, from that point forward, it’s as though you were rolling a die with that many sides (since you ignore any higher numbers you might come across). This enabled him to set up what’s known as a “recurrence relation,” an equation that tells you later values in a sequence in terms of earlier values in the sequence.

It helps if we define SN as the average score you’ll get when playing with a die numbered from 0 to N. So if you’re playing with die numbered from 0 to 9, your average score will be S9. Another way to compute this average is to break it down into 10 cases, depending on what your first roll is. If your first roll is a 9, then your average score will be 0.9 plus the average of all the scoring you’ll do thereafter. Because all the numbers are still in play, the only thing that’s different now is that your scoring has moved to the right by one decimal place, meaning your rolls now count for one-tenth of what they used to. In other words, when your first roll is a 9, your average score will by 0.9 + 0.1S9.

And what happens if your first roll is an 8? Your average score will be 0.8 plus the average of your scoring thereafter, which turns out to be 0.1S8. Continuing all the way down to S0, the resulting equation for S9 is — take a deep breath — S9 = 0.1(0.9 + 0.1S9) + 0.1(0.8 + 0.1S8) + 0.1(0.7 + 0.1S7) + 0.1(0.6 + 0.1S6) + 0.1(0.5 + 0.1S5) + 0.1(0.4 + 0.1S4) + 0.1(0.3 + 0.1S3) + 0.1(0.2 + 0.1S2) + 0.1(0.1 + 0.1S1) + 0.1(0 + 0.1S0). The coefficient of 0.1 in front of each term comes from the fact that each case — rolling a 9 first, rolling an 8 first, etc. — has an equal probability of one-tenth.

What a recursive mess. What if we were to look at smaller values of N instead? Well, we know S0 = 0, since it represents your average when all you can roll is a 0. The equation for S1, the average score when you can only roll a 0 or a 1, is slightly more complicated: S1 = 0.5(0.1 + 0.1S1) + 0.5(0 + 0.1S0). Using the fact that S0 = 0, we can solve this equation for S1, finding that it equals 1/19. Next, we can use these values for S0 and S1 to find S2, which turns out to be 2/19. Lo and behold, a beautiful pattern emerges: SN equals N/19. That means S9 equals 9/19, our answer. Honestly, it’s pretty satisfying to see that in this game, where all the scores are decimals between 0 and 1, the average score winds up being fairly close to 0.5.

Meanwhile, solver Ignas from London took a different approach, noting that your first roll can be anywhere from 0 to 9 with equal probability, so the average value of the digit in the tenths place will be 4.5 (halfway between 0 and 9). Whatever your first roll was, your second roll is constrained to be between 0 and that first roll, so the average value of your second roll will again be halfway between 0 and your roll. This also means that the average digit in the hundredths place will be 2.25 (halfway between 0 and 4.5), the average digit in the thousandths place will be 1.125 (halfway between 0 and 2.25), and so on. Summing the resulting geometric series again gives an average final score of 9/19.

The puzzle’s author, Ricky Jacobson, also found the average final score in the general case of an k-sided die: (k−1)/(2k−1).

Finally, as I had hoped, several members of Riddler Nation turned to their computers for help, simulating loads and loads of games. Robert DiMartino kindly provided a link to his code that will run 10,000 games in under a second. Solver Angelos Tzelepis went so far as to play 3 million games, finding an average score of 0.47356, within spitting distance of 9/19. He even achieved a top score of 0.99999995. Impressive!

Solver Hernando Cortina plotted a histogram of his scores, revealing which are more likely, less likely and downright impossible:

Think that’s cool? It turns out that graph is actually fractal! As you zoom in on scores closer to 1, the pattern repeats itself. Crazy, I know.

Want more riddles?

Well, aren’t you lucky? There’s a whole book full of the best puzzles from this column and some never-before-seen head-scratchers. It’s called “The Riddler,” and it’s in stores now!

Want to submit a riddle?

Email Zach Wissner-Gross at [email protected].

Significant Digits for Friday, Nov. 22, 2019

You’re reading Significant Digits, a daily digest of the numbers tucked inside the news.


$26 million in missing paychecks

Millions of people around the world depend on automatic payroll deposits being regularly delivered into bank accounts to do things like pay rent, buy groceries, and make student loan payments. That didn’t stop the president of a New York payroll management company called MyPayrollHR from redirecting $26 million in payroll funds directly into his personal accounts. It wasn’t his only major instance of financial redirection. After being arrested and charged with bank fraud, former MyPayrollHR president Michael Mann admitted in court to stealing almost $70 million over the last nine years. [New York Times]


1,350 women with defective medical devices

It took seven years and a class action suit involving more than 1,350 women, but an Australian Federal Court judge has found a subsidiary of Johnson and Johnson was actively negligent and is now liable to compensate patients who suffered injuries from defective permanent medical devices. Women involved in the class action suit had transvaginal meshes inserted into their bodies, but many of the implants resulted in severe complications among patients, including incontinence and chronic pain. [BuzzFeed News]


2,400 layoffs

The bad news continues for office-sharing company WeWork, after the company announced it was laying off 2,400 employees on Thursday. The staffing measure is an effort to cut costs and refocus on the office-sharing component of the business. The company had expanded to several other side ventures such as a school and co-living facilities, but that’s all crumbling now. After the company pulled back its IPO filing in September, co-founder Adam Neumann stepped down from his role as CEO after intense scrutiny over the organization’s bad balance sheet. [CNBC]


1 ice cream spoon

Discarded items, even small ones from a popular ice-cream chain, can hold important biological clues to crimes that happened decades ago. This week, officials in California announced they had identified a suspect in two cases of sexual assault from 1997 by using DNA evidence that had been collected from a Baskin-Robbins spoon. A 60-year-old man has now had multiple felony sexual assault charges filed against him. [Los Angeles Times]


20 years old

When I was 20 years old, I was still an undergrad, trying to figure out what I wanted to do with my life. At 20, Luka Dončić is already playing for the Dallas Mavericks, and averaging mind-numbing numbers: 29.9 points per game, 10.6 rebounds, and 9.4 assists this season in the NBA. FiveThirtyEight’s Jared Dubin points out the only other player averaging similarly high numbers is LeBron James. Dubin also notes that Dončić is among the youngest players in the league’s history to achieve these statistics and is “well on his way to the best age-20 season of all time.” [FiveThirtyEight]


145 graves

The location of a high school in Tampa, Florida will likely be moved after the graves of 145 people were discovered underneath the building. The graves are part of Ridgewood Cemetery, a burial location for the poor that was originally owned by the city, then sold off to a private company before being acquired by the school district in 1959. The school district said records show more than 250 people were buried at Ridgewood, most of them African-Americans, and up to a third of the burials were for babies and small children. [BBC News]


SigDigs: Nov. 22, 2019

The 2020 Front-Runners Aren’t As Well-Liked As Past Contenders

Welcome to Pollapalooza, our weekly polling roundup.

Poll(s) of the week

We know that former Vice President Joe Biden, Sen. Bernie Sanders and Sen. Elizabeth Warren are popular among Democratic voters. But how do they fare among the general public? A recent poll from Gallup shows that opinions about this trio are far more mixed among all Americans than they are among Democrats. And relative to previous high-profile candidates, they don’t seem to be as popular. Gallup’s poll found that the net favorability (favorable rating minus unfavorable rating) of these leading Democrats was roughly even or negative among the general electorate, with Sanders at +1, Biden at -1 and Warren at -5. Granted, we found that none of them are as unpopular as Hillary Clinton or Donald Trump were at this point in the 2016 cycle, but they’re also not as favorably viewed as George W. Bush in 2000, Barack Obama in 2008, or even Rudy Giuliani in 2008.

We looked at non-incumbent candidates’ favorability ratings in each cycle dating back to 2000, focusing on the last four months of the year before the election. We found that Biden, Sanders and Warren are less well-liked than many well-known presidential contenders from the past two decades.10 These numbers could be a red flag for Democrats looking ahead to the general election or a product of our increasingly polarized politics. You can see that since 2008, many candidates’ net favorability ratings have been negative or close to zero.

Biden, Sanders, Warren aren’t as popular as past candidates

Average net favorability ratings (favorable rating average minus unfavorable rating average) of well-known presidential candidates* in national polls from the last four months of the year preceding the primary, 2000 to 2020

Candidate cycle party Fav Avg Unfav Avg Net
George W. Bush 2000 R 56.2% 23.4% +32.7
Barack Obama 2008 D 49.3 29.8 +19.5
John McCain 2008 R 45.3 31.2 +14.0
John Edwards 2008 D 45.5 32.5 +13.0
Rudy Giuliani 2008 R 47.5 35.2 +12.2
Al Gore 2000 D 43.2 39.0 +4.2
Hillary Clinton 2008 D 47.5 45.5 +2.0
Mitt Romney 2012 R 37.5 35.8 +1.7
Elizabeth Warren 2020 D 39.8 40.2 -0.4
Joe Biden 2020 D 43.4 44.3 -0.8
Bernie Sanders 2020 D 43.6 45.2 -1.6
Hillary Clinton 2016 D 41.9 50.9 -9.0
Newt Gingrich 2012 R 30.8 45.3 -14.5
Jeb Bush 2016 R 30.8 51.1 -20.3
Donald Trump 2016 R 32.6 58.7 -26.1

Data does not include incumbent presidents. Additionally, we only included candidates for whom at least 75 percent of respondents had an opinion (the lone exception is Mitt Romney, who was within two points of the threshold, but as the 2012 Republican nominee, we included him). In 2004, no Democratic candidate was close to clearing the 75 percent threshold. When a poll presented multiple observations, larger sample populations were prioritized, such as adults over registered voters or registered voters over likely voters.

*Data for 2020 candidates is from polls released through Nov. 20, 2019.

Source: Polls

By comparison, the top-tier candidates in the 2000 and 2008 campaigns all had net positive favorability ratings at this point — some of them quite high, too. Bush, for instance, had by far the strongest numbers of any candidate, at +33 points. Obama, John McCain, John Edwards and Giuliani also had net favorability ratings higher than +10. Al Gore and Clinton11 were closer to an even net favorability rating, but they were still viewed somewhat positively.

But a candidate’s favorability ratings at this point don’t necessarily line up with election results — a lot can change between now and next November. For instance, while Bush was viewed more favorably than Gore in late 1999, they fought out the 2000 election to a near-draw, which in the end was decided in Bush’s favor by an incredibly narrow margin in Florida. And in 2016, Trump won the presidency over Clinton despite being viewed less favorably, which remained true through Election Day.

The good news for Democrats is that Americans like Trump even less. Gallup’s poll found Trump’s net favorability at -18, far below the three leading Democratic contenders. So, in other words, as long as the Democratic nominee wins over those who view the president negatively, even an unpopular nominee could still have a pretty good shot at winning. Still, Democrats could find themselves in trouble if the election becomes a race to the bottom, where both Trump and the Democratic nominee are heavily disliked. Exit polls in 2016 found that Trump still won 15 percent of voters who had an unfavorable opinion of him, as he was likely aided by the fact that Clinton was also viewed pretty negatively.

Other polling bites

  • A survey by The Washington Post and the Kaiser Family Foundation found that Democrats were more likely than Republicans to connect extreme weather patterns to climate change. Nationally, the widest disagreement between the two groups was over whether climate change played a “major factor” in the occurrence of extremely hot days. Seventy percent of Democrats said yes, compared with only 24 percent of Republicans. There were also regional disagreements, as 63 percent of Democrats in California and the Southwest said climate change was a major part of droughts and water shortages, versus 20 percent of Republicans in those states.
  • New polling from the Institute of Politics at Harvard University found that 18-to-29 year olds likely to vote in 2020 are divided over whether they prefer big, sweeping change versus a slower, more pragmatic approach to governing. Among those likely to vote in the general election, 44 percent said they prefer policies “that stand a good chance of being achieved,” while 40 percent want major structural changes “even if they will not be easy to carry out.” But among those planning to vote in the Democratic primary, this result was flipped: Forty-five percent said they prefer sweeping changes, versus 39 percent who said they prefer more achievable goals.
  • In last week’s Pollapalooza, we looked at former New York City Mayor Michael Bloomberg’s nascent presidential campaign and found that his horse-race and favorability polling weren’t that great. A new national survey of Democrats from Reuters/Ipsos released this week echoed this analysis, finding Bloomberg at 4 percent. That poll also found that only 7 percent of Democrats viewed Bloomberg “very favorably” compared to 26 percent or more for candidates such as Biden, Sanders and Warren. Notably, however, Bloomberg’s limited support seemed to come at the expense of Biden, who fell from 30 percent to 26 percent when Bloomberg was added as an option.
  • A new study from Gallup and the Knight Foundation found that 56 percent of Americans believe local news organizations are doing “very well” or “somewhat well” financially, at odds with the difficult reality facing these outlets. However, Americans still value local news — 86 percent said everyone should have access to it even if they don’t pay for it — and education about the trying media landscape and journalism’s role in a democratic society could help attract more financial backing. When supplied with information about the financial problems faced by local media and journalism’s positive effects on democracy, 58 percent of respondents said they would be willing to donate to a local news nonprofit organization versus only 40 percent among those who weren’t told this information.
  • New data from YouGov shows that 32 percent of parents with children under the age of 18 believe that the flu vaccine can actually cause the illness itself, which is a myth the Centers for Disease Control and Prevention has worked to debunk. However, despite misconceptions surrounding the flu vaccine, 60 percent of adults said they plan to get one this year.
  • The Public Religion Research Institute and AAPI Data collaborated to survey over 2,500 Asian Americans and Pacific Islanders (AAPIs) in California, and what they found was a state of “two Californias” among AAPIs, with one group enjoying financial stability while the other is financially insecure. In terms of economic security, 23 percent of AAPIs reported they were working but struggling with poverty, compared to 37 percent who said they were working and not struggling (another 40 percent were retired, students or not otherwise working). But 62 percent of AAPIs believed the “American Dream” still holds true, though this was much higher among those who had immigrated to the U.S. (69 percent) compared to native-born AAPIs (43 percent).
  • The cost of health care in the U.S. has gotten a lot of attention in the 2020 campaign, and a new study from Gallup and West Health found that roughly 13 percent of Americans reported knowing at least one friend or family member who died in the past five years after not receiving medical treatment because they couldn’t afford it. Another 23 percent of Americans said that at least once in the past 12 months they or someone in their household couldn’t afford the medicine or drugs prescribed to them. Sixty-nine percent of Americans said the costs of prescription drugs are “usually much higher than what consumers should be paying,” though Democrats were somewhat more likely than Republicans to think this (76 percent to 64 percent).
  • Tuesday marked the first debate ahead of the United Kingdom’s general election, and a post-debate survey of debate viewers by YouGov found that 51 percent felt Prime Minister Boris Johnson of the Conservative Party won, compared with 49 percent for Labour Party Leader Jeremy Corbyn. The Economist’s polling tracker shows Conservatives holding a 43 percent to 30 percent lead over Labour.
  • Thanksgiving is almost here, which means many Americans can look forward to arguing over divisive political topics with family members. But on a lighter note, YouGov delved into a classic Thanksgiving debate by asking about turkey meat preferences. Apparently ignoring their taste buds, 50 percent of respondents said they prefer eating white meat, while only 32 percent preferred dark meat.

Trump approval

According to FiveThirtyEight’s presidential approval tracker, 41.9 percent of Americans approve of the job Trump is doing as president, while 53.6 percent disapprove (a net approval rating of -11.7 points). At this time last week, 41.2 percent approved and 54.5 percent disapproved (for a net approval rating of -13.3 points). One month ago, Trump had an approval rating of 41.2 percent and a disapproval rating of 54.5 percent, for a net approval rating of -13.3 points.

Generic ballot

In our average of polls of the generic congressional ballot, Democrats currently lead by 5.8 percentage points (46.8 percent to 41 percent). A week ago, Democrats led Republicans by 5.7 points (46.8 percent to 41.1 percent). At this time last month, voters preferred Democrats by 6.3 points (46.6 percent to 40.3 percent).