There was a lot of unwarranted buzz this past week about the New England Patriots winning 19 of their past 25 coin tosses prior to each game, leading to semi-serious speculation of more foul play. This episode is a great source of many interesting probability problems, none of which is the question asked by almost every related news article: “What is the probability that the Patriots would win at least 19 of their last 25 coin tosses purely by chance?” (The answer is about 1/137, but the better answer is that this is the wrong question.)

I think one slightly better and more interesting question is: how *often* does this happen by chance? That is, suppose that we flip a fair coin repeatedly, and after each flip we observe whether at least 19 of the *most recent* 25 flips came up heads (a success), or less than 19 were heads (a failure). Over time, for what fraction of flips should we expect success?

Another related question: what is the expected number of games until the *first* such success?

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Amortized, isn’t that the same as the first question? (Though, ~2/137 in this case since both “>= 19” and “<= 6" are being considered.)

I see now that my language wasn’t very clear; the intent was to count only “>=19” as a success, and to ask what fraction of observations would be successes, *not* “failures” of “<=6".

At any rate, you're right– the first was a bit of a trick question, since the expected fraction is indeed the same as the probability for any particular sequence of 25 flips. It's the follow-on that is different, and (I think) significantly more difficult.