Tuesday, November 17, 2015

Half-time scores

This discussion was suggested by a commenter who was asked this question in a math class, presumably as a "real world" word problem. These are just some rough notes relating to a string of conversations we've been having around this idea.

The scene
Our friend calls excitedly to tell us about the football (aka soccer) game she just saw: "[favorite team] was leading 2-1 at halftime." Suddenly her phone runs out of batteries and we don't get the final result. Can we figure out what it was?

A math class answer

4 to 2 victory for our side, of course. This is the naive model where the scoring rate is constant over the course of the game, so 2x as much time means 2x the score. Nonsense, for anyone who has a passing familiarity with the real game.

In discussion, one of the J's offered 3 vs 3 as an alternative and explained his thinking was that they change ends at half-time. The underlying model was that the direction of play determined the outcome. Quite a strange model!

Some stats

Given a half-time score, what can we say about the final result?
  1. scores go up, so the interim score for each team is the least they could have at the end
  2. Intuitively, the team leading at half-time is likely to win the game
There is some intriguing data related to half-time and full-time scores on the OptaPro Blog and they have a further link to the Football Observatory. One thing we saw right away is that 2 -1 and 1-2 halftime scores are fairly uncommon (about 5% of the sample games, when taken together). Perhaps this is why 2-1 and 1-2 halftime scores weren't included in some of their conditional tables, though, together, that was still about 970 matches (17,656 * 5.5%).

Corroborating our intuition from point 2, we looked at the 1-0 and 0-1 lines in their table 6 to guess that 2-1/1-2 matches would also have roughly a 30% chance of a change in outcome. The table doesn't specify whether the change is to a tie or a change of winner, but we guessed that the latter was less than half the change cases.

Comparing with other sports

The last topic we discussed was to compare with other sports, for example professional basketball.
First, if we kept the scores unchanged (2 vs 1 at half-time), then we know that we are watching an extremely unusual basketball game. At that point, we are so far into the tail of the distribution that it is hard to know what is happening and very dangerous to make guesses about the rest of the game.

Second, let's say that we have a more reasonable half-time score, but one team leads by a single point. We aren't so familiar with NBA results and I couldn't find a great stats source, but we assumed one team had 51 points and the other 50. In contrast with football, we concluded that this was not likely to tell us much at all about which team would win at the final whistle.

What if, instead, we assume a 60-30 point split? Well, in this case, we reasoned that a guess of 120-60 was much more reasonable because each scoring event is much small and more frequent than in football. Also, we were much more sure that the leading team at half-time had demonstrated  statistically significant strength relative to the trailing team. We were pretty confident that they would win in the end.

However, we also recognized that confidence was not mathematical certainty. Even in basketball, scoring doesn't happen at a continuous rate. Also, it was easy for us to come up with events (player substitution, player injury, change of strategy, fatigue) that would create a different scoring rate in the second half.

Your turn

What about you? Have any favorite "real world" questions from math class that, when you use your own real world experience, are actually very silly? Any beloved sports which you think offer another point of comparison for our discussions? Maybe games like cricket, baseball, or tennis where the end of the game is not determined by time?

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