The ESPN app and ESPN.com include a feature that allows you to see the probability, at any point in a basketball game, that a team will win. Based on these stats, the Boston Celtics appear to be defying the odds on a regular basis. Are they really this great, or is it just a statistical paradox?

The Celtics are an incredible story this year. New players, including rookie Jayson Tatum, account for 54% of the team’s minutes. One of the team’s best players went down with a season-ending injury 6 minutes into the first game. And yet the team has played together incredible well, and has ripped off a 16-game winning streak including some unlikely comebacks.

How unlikely?

ESPN’s “win probability” feature purports to answer that question. Here’s a chart from one game that featured an amazing turnaround, at the Dallas Mavericks on November 20.

Here’s what this says. The line on the chart shows the probability at any point in the game that the Celtics or Mavericks will win. With 7:35 left in the fourth quarter, Boston is down by 13 points, 87 to 74, and Dallas player Harrison Barnes has just grabbed the rebound on a Celtic missed shot. According to ESPN, the chance of the Mavericks winning this game at that point was 98%.

The Celtics defied the odds and won in overtime, 110 to 102. That was a 50-to-one shot.

That’s not the only one. Here they are against the defending NBA Champion Golden State Warriors on November 16.

With just under 5 minutes left in the third quarter, Boston’s Kryie Irving has just fumbled a pass and last year’s best team is ahead by 17, 66-49. ESPN estimates it’s a 97% chance that Golden State will win, and any rational person would agree. But the Celtics come back to beat them, 92-88.

There’s more. Here they are against the Charlotte Hornets, a game in which their best player Irving got an elbow to the face partway through. Down 76-62 at the end of the third quarter, ESPN rated their chances of losing at 93%, without even accounting for Irving’s cracked skull, but they come back anyway.

And against Oklahoma City, just after the half and down by 18 points, ESPN says there’s a 96% chance the Thunder will win. They lose.

What’s going on here? There are three possible explanations.

## Explanation 1: The Celtics have defied the laws of probability

These charts reflect four events — four points in these games when the Celtics could either win or lose. The chances of them winning all four, if you trust ESPN’s probabilities, are 2% x 3% x 7% x 4%. That’s a probability of 0.00000168, or one chance in 600,000.

To say that these Celtics are extremely good at comebacks is an understatement. (They’ve of course cruised to some easier wins and lost a few games since the streak ended, but it’s the comebacks that stand out.)

But one in 600,000 is an extraordinarily unlikely outcome. Their coach is well suited to the team. They’ve had some luck, too. They’re confident. But one in 600,000 — is that possible without divine intervention?

## Explanation 2: ESPN’s probabilities are inaccurate

Can we trust ESPN’s numbers? While it hasn’t published its algorithm, we can make a guess. You can look back at a sample of thousands and thousands of NBA games. Each of those games has hundreds of moments in it. As a result, you can ask a question like “When the home team is down by 16 with 5 minutes left in the game, how often do they win?” I’m sure that the algorithm is based on something like that.

You can see a similar estimation tool from inpredictable.com here. For example, I plugged in the data about the 13-point deficit, 7:35 remaining, and the other team in possession of the ball from the Celtics-Mavericks game in my first example. Inpredictable said there was a 96% chance the team that was behind would lose, similarly to what ESPN predicted.

There are also the teams’ abilities. I looked at the Celtics’ most recent game, a loss at home against the Detroit Pistons a couple of days ago. At the very start of the game, ESPN estimated that there was a 75% chance the Celtics would win. This tells me that ESPN also factors in the record of the teams playing (Boston was 18-3 and Detroit was 12-6) and the home court advantage.

Based on this information, I think ESPN’s estimates are sound. Until you can look into the hearts of the players, this is the best estimate you’re going to get. So perhaps the Celtics really are having an unprecedentedly unlikely run.

## Explanation 3: We are fooling ourselves about the probabilities

Do one-in-a-million things happen? If you look at a million instances, sure.

The one-in-600,000 chance I cited earlier was picked to make a point. How many instances of ESPN estimates are there altogether?

Each team plays 82 games. There are 30 teams. So there are 1,230 regular-season games in an NBA season (since there are two teams in each game).

How many of these “moments” are there that ESPN can estimate the probability for? Each team has about 100 possessions per game. Each possession is an event. Each team commits about 20 fouls per game, gets about 50 rebounds, and commits about 15 turnovers. That’s 185 events per team, or about 370 per game. There is some overlap, but let’s just guesstimate that there are around 370 moments in an NBA game on which the ESPN estimator could make an estimate.

That means there are 455,000 moments to estimate.

I have no idea how many of those estimates are 95% win probability or higher. But there are at least 110,000 of those estimates in the fourth quarter. Would it be unreasonable to believe that of those 110,000 instances, 20,000 involve leads large enough for at least a 95% win probability?

That’s a wild guess, but it means that 5% of those 20,000 instances, or 1,000 cases, will see the underdog come back and win it. I found four involving the Celtics.

If I am right about this, I should be able to find more, from other teams. And I did. It was easy. Here are the Trail Blazers coming back from 17 points down to turn the tables on the Washington Wizards last Saturday. ESPN said there were 99.3 chances out of hundred that wouldn’t happen — it was a 143-to-one shot that the Trail Blazers would win. But they did.

There are lots of games just like this, with just as unlikely outcomes. The reason is that the sample is so large.

There are three lessons here.

One is that lots of NBA games are pretty damn exciting, and the unlikely happens all the time. If you like unlikely comebacks, somewhere in the NBA on any given night, one is probably happening.

The second is that our intuition about probability is off. Unlikely events happen all the time, even one-in-600,000 shots. You just need a large enough sample to see them. Since they stand out, we notice them.

And the third is that no matter how you slice it, these Celtics are pretty damn amazing. Unlikely things may happen all the time, but when they happen often enough to one team, you have to give that team credit.

Obviously, my estimates in this section are rough, but the principle is sound, I think. If you’re a stats wizard, I’d welcome your additional insight.

Interesting, Josh. I did something similar with hoops a few years ago with the Warriors:

https://www.philsimon.com/blog/data/big-data/what-are-the-odds-that-the-warriors-go-4-of-30-again/

This is a pisstake on the bullshit poured out by fans of minority sports, right? (Basketball is followed by less than one in twenty Australians.)