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?