## What is the chance of an NBA comeback when down by 35?

By Professor Carl Morris

The Kings mounted the second largest comeback in NBA history.

The Sacramento Kings came back to win on Monday night (Dec 21), when down by 35 points against the Bulls in Chicago with 20 minutes and 50 seconds (20:50) left.  When two even NBA teams play the chance that the trailing team would win from that situation is about one in 24,000.  Let’s take an in-depth look at the calculation…

Any such calculation requires many assumptions.  These chances can be made larger or smaller by changing the evenness of the teams, their scoring rate, and so on.  The calculations here assume that the teams score at the usual rate per game (about 100 points each per full game) and that the standard deviation of the scoring margin is about 13.5 points in a game (no exact value is possible).  A larger standard deviation would improve the chances of the trailing team.

Historically, well over 24,000 NBA games have been played, but in only a small percentage of those does a team trail by 35.  Usually the trailing team would be the noticeably weaker team with a much lower comeback chance than calculated here.  To allow time for a comeback, any large (e.g. 35-point) difference must occur with substantial time remaining.   For example, with 12 minutes to play a 35-point comeback has a prohibitive one chance in 10 million, while with 30 minutes remaining the chance rises to one in 2000.  The Elias Sports Bureau lists the largest ever comeback as 36 points, in a 1997 game, but that was in the first half and makes that (still very unlikely) comeback several times more likely than in Monday night’s game.

On Monday, Sacramento had the advantage of possession with 20:50 to go. Accounting for this, and that Sacramento would be favored by 2 points (based on the teams’ past records) over a 20:50 minute period, the chance of their winning from 35 down rises to about 1/8000.

The theory used for this calculation is based on independence, while streaks and psychology actually can create (positive) dependence.  When that happens, it increases sigma (aiding the trailing team, assuming the dependence is could affect either team).  Three point opportunities assist teams that trail.  Sacramento attempted many threes in the second half, making them with regularity.  Chicago may have been put on their heels by adopting a “prevent defense” strategy (an apt term in that so often it prevents defense). Chicago had just 9 FG attempts in the 4th quarter, 2 of those when they trailed at the end.

Ignoring end-game issues, coming back to win rises in relation to the magnitude of the lead, divided by square root of the remaining time (based on a Brownian motion model).

Accounting for end-game situations and 3-point basket possibilities, it seems better yet to propose a comeback index of:

(lead minus 3) / sqrt(time remaining).

Monday night’s index calculates to  (35 – 3)/ sqrt(20.83 min)  = 6.46.  I have no idea as to whether 6.46 is the record for this index.  A further correction could/should deduct another half point to account for possession.

Bill James’ formula for NCAA basketball states that a 35 point lead without the ball and with 21 min to play is “79% safe”.  The definition there of “safe lead” clearly understates how amazing a comeback like Sacramento’s was.  A better definition would be based on the probability of winning, which for Chicago was way over 99.9%.