Every year there’s a Super Bowl, and every year the whole shebang gets started by a famous person tossing a coin into the air. The team winning the toss gets to decide whether they want to begin the game on offense or defense. Theoretically this choice might produce an advantage. If so, would be interesting to know how much. The same thing happens in physics – just how much signal is hidden in the random noise of an experimental apparatus?
Let’s take a look at the numbers and try to see what kind of advantage the toss-winning team has. The data is pretty straightforward – in 43 coin tosses, the winner has gone on to win 20 games and lose 23 games. As such it would seem that winning the toss is a disadvantage. But is it, or is the seeming disadvantage just a matter of the random fluctuations inherent in a small sample size? Keep in mind that the fairness of the toss itself isn’t really the issue. If they all came up heads that wouldn’t necessarily have any bearing on whether winning that biased toss was actually an advantage in winning the game. We just want to know if a toss win has any bearing on game outcome.
If there were no relationship between wining the toss and the game, we can expect that the probability of a game win is 0.5 given a toss win. Now it’s time to break out some math: since the sample size n = 43 and probability p = 0.5, the normal distribution of mean np and variance np(1-p)is a good approximation for the distribution of game wins that the toss winner ought to experience.
The standard deviation is the square root of the variance – in this case, the standard deviation is 3.28. In the normal distribution, about 68% of the time the actual number of successes is within one standard deviation of the mean. Here, the mean is 21.5 game wins. Therefore there’s a very good chance (~32%) that statistical fluctuations will put the actual number of game wins outside that range without the toss giving an actual advantage or disadvantage. Thus if the number of game wins is inside the range 18-25, we have no grounds for concluding that victory in the toss affects victory in the game. Since the actual number of game wins is 20 – well within that range – it’s not especially likely that you need to worry if your team loses the toss. If the actual number of game wins were outside the one-standard-deviation interval but within the two-standard-deviation interval (about 14-28) we might raise an eyebrow, but we still wouldn’t necessarily be on very solid ground to assume a relationship. Outside the two-standard-deviation interval we might be justified in suspecting a relationship between toss wins and game wins. A larger n would help clarify the issue, as it shrinks the standard deviation compared to the expected number of wins.
The coin toss that determines possession in overtime is another story. There the number of game wins compared to toss wins is wildly out of proportion to the p = 0.5 hypothesis, and the winner does in fact have an important advantage. But that’s a story for another time.
[Statisticians will justly note that what I have not strictly done is to calculate estimated probability of a game win given a toss win. This is true, but is also more complicated and doesn't accomplish much that we didn't already learn by working in terms of deviation from an assumed p = 0.5.]