Puzzle: odds of a comeback victory

You’re favorite sports team is down at halftime, but you are hoping they can pull it together and eek out a victory. What are the odds of that happening?

The probability obviously depends on the exact game in question. But I came across an interesting probability problem that gives an interesting insight on the odds of a comeback victory.

Here is the problem:

Consider two teams A and B that are completely evenly matched. Given that a team is behind in score at halftime, what is the probability that the team will overcome the deficit and win the game?

Assume there are no ties, and the result of the first half does not affect how players perform in the second half (that is, the first and second half are taken to be independent events).

(problem based on page 11, “Probability: the language of randomness,” by Jeffry S. Simonoff)

While the assumptions of this problem may seem unrealistic, I will point out that actual data from NBA and NFL games on a league level for certain years has almost exactly matched up with the calculated probability (more about this in the solution).

So I still think this problem has some real world applicability even though it is a simple and fun model.

Can you figure it out?

As usual, the answer is in the comments section.



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  • http://www.mindyourdecisions.com/blog/ Presh Talwalkar

    Puzzle solution:

    Because the teams are evenly matched, you might mistakenly think the answer is 50 percent. But that is the probability the team would win overall. If a team is down at half-time, the chances of winning will be less. So let us try to calculate the odds.

    We have to think about how a team could have a comeback victory if it is down at halftime.

    Let us first write down the possible outcomes of the game, broken down by halves. Since the two teams are evenly matched, there are four different possibilities for who is leading during each half (ignore the case of a tie):

    (first half, second half):
    AA
    AB
    BA
    BB

    Because the teams are evenly matched, these events are all equally likely so each occurs with probability 1/4 = 25 percent

    In two of the cases, one team scores more points in both halfs of the game, and there is no come from behind victory: AA and BB. This means 50 percent of the games the team that lags behind at half ends up losing the game.

    The other two possibilities are times when a team could have a comeback victory. In these cases, one team leads at the half, but gets outscored by the other in the second half: AB and BA. In order for a team to get a comeback victory, it must overcome the deficit from the first half. How often does that happen?

    The answer can be calculated by the following logic: since the two teams are evenly matched, it is equally likely that the team will score enough points to overcome the deficit or that it will not score enough points. (For instance, the event of falling behind 6 points in one half happens with the same probability of gaining 6 points in a half). Therefore, in the event AB, it will be equally likely that B scores enough to eventually win, or that it would not score enough and it loses.

    Therefore, B ends up winning in half of the cases, or 12.5 percent of the time (take 1/2 of 25 percent). The same logic applies for the event BA: there is a 12.5 percent chance that team A ends up winning.

    Putting this all together, we have:

    Probability(team having comeback victory) = P(AB)*Pr(B wins) + Pr(BA)*Pr(A wins) = 12.5 + 12.5 = 25 percent

    So under these assumptions, a team will have a 1 in 4 chance of making a comeback victory.

    Now you may point out this is not realistic as the model does not take into account quality of teams and things like home field advantage. Nor does it take into account psychology: a recent study shows that teams with a slight deficit at halftime end up winning more often than teams with a slight edge at halftime. Here is the remarkable study based on 18,000 professional basketball games and 45,000 college games.

    However, even though the assumptions are a bit off, the overall league statistics seem to mirror the probability model.

    In the National Football League, a small sample of games in 2005 showed this trend:

    Joe Gibbs is not telling his troops they have a 23 percent chance of winning. Of the 88 games observed, 68 of the teams that went in at halftime with the lead went back to the locker room at the end of the game with the lead and the win. That’s right 77 percent of the time if a team had a lead at halftime, it won the game. [And thus 23 percent of the time, the team facing a deficit came back for a victory]

    I found the same pattern was shown to happen in the National Basketball League (though granted this is 20 year old data; I’d love to see whether the pattern holds true for more recent seasons):

    Professor Hal Stern of the University of California at Irvine examined 493 National Basketball Association games from January to April 1992, and found that in 74.8% of the games, the team that was ahead at halftime ultimately won the game [and thus the losing team at halftime came back with probability 25.2 percent]

    This is either a pure coincidence or there is something to be said about the simple probability model. It’s fascinating to me either way.

  • john

    I think the statement of your problem is a little ambiguous. For a specific game the probability of a comeback depends on the score at the end of the first half. The bigger the spread, the smaller the chance of a comeback.

    But at the beginning of the game (or at half time if you don’t know the score), there is a 25% chance of a comeback.

    Seeing this is actually fairly simple, there is a 50% chance for each team to win the second half. If it’s the team that one the first half, there is no comeback. If each team wins one half (this happens the other half of the time) the spread of the second half will be greater half the time and the spread of the first half will be greater half the time. A comeback occurs when and only when the spread of the second half is greater. So 1/2*1/2=1/4 of the time.

    It seems quite surprising that teams that aren’t evenly matched would show this same ratio (one would expect fewer comebacks). I guess the psychology compensates? Are better teams more likely to be careless in the first half? And your study of 18000 pro, and 45000 colege games does suggest that the leaders at half time don’t put as much effort into the second.





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