I'll put all correct solutions received by MONDAY September 10 into a hat, and draw a winner, who will get a small prize.
The puzzle: I have two coins in my pocket. One is fair: it comes up heads half the time. The other is biased in favour of heads: it comes up heads with probability .7 each time I toss it. I can't tell the two coins apart.
I pick one coin from my pocket. Clearly the probability that it is the fair coin is .5. But then, I toss it four times in a row, and each time I get a head. Clearly this should change my assessment of the probability that I am holding the fair coin - it should make that probability somewhat less than .5. But how much less?
So here's the puzzle: what's the probability that I picked the fair coin, given that when I tossed it four times in a row, it came up heads each time?
And what's the answer when I replace "four" with "n", for an arbitrary whole number n?
Solutions, together with the list of correct solvers and the prize winner, are here.