Citing a CBS article, Uncommon Descent complains “No design inference allowed on coin flips.” As I’ve come to expect from them, it’s pretty hard to parse a coherent argument out of the article, but I’ll bet it has something to do with 747s and tornadoes (say what you want about Evolution News and Views; at least you can follow their arguments). So it’s not clear what design inference they think we should draw from the New England Patriots winning 19 of 25 coin flips, nor who is being prevented from drawing it.
Leaving aside the bad evolution analogy, the CBS article claims that
…the probability of winning it at least 19 out of 25 times is 0.0073 — less than three-quarters of one percent!
This is true, but irrelevant. The probability of this happening is the probability of 19 wins plus that of 20 wins, plus that of 21 and so on up to 25. As a binomial probability with a probability of success of 0.5 on each trial, this is
which is indeed about 0.0073. But why are we focused on this particular team and these particular 25 coin flips? Specifically because they are unusually far from the expected outcome. So now we have to ask what other outcomes would have caught our notice. Certainly 19 Patriot wins in ANY 25 consecutive coin flips would do it. The Patriots have been playing for 55 years, around 14-16 games per year, not counting post-season games, so let’s call that 825 games. Even if we split that up into blocks of 25 games (which is way overly conservative), there are 33 such blocks. The probability that any one of those blocks (their first 25 games, their second 25 games, and so on) would include 19 or more winning coin flips is around one in four. Not all that surprising. If we allowed any consecutive 25 games (say, games 12-36), the probability is even higher.
But we would surely be having a similar conversation if the Steelers had won 19 of 25 coin flips. Or the Bills. Or the Seahawks; in fact, any of the 32 NFL teams. Not all of these teams have been around as long as the Patriots, so the math gets, well, more involved than I have time for. We’d be having a similar conversation if any of those teams had won 10 coin flips in a row (probability about 0.001 per team per ten games). Or 14 out of 15 (0.0005). Or if a high-profile college team had had a similar run. If a team had lost 19 of 25 coin flips, we’d be wondering where their bad luck came from.
The problem with all of these conversations, real and imagined, is that they fail to consider the PETWHAC, as Richard Dawkins calls it: the Population of Events That Would Have Appeared Coincidental. In all of the thousands of football games that have taken place in the last half-century, a run of 19 winning coin flips out of 25 is a near certainty. By focusing on a particular outcome because it is unusual, all we are doing is anomaly hunting. In any large sample, there are likely to be anomalies; we should only be surprised if there aren’t!