Re: fun probability question from an old Putnam
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Initial guess: very very low.
Reason: the distribution will be bi-modal with lots of high and low totals because as soon as he gets above or below 50% (most noteably after the third shot) his precentage will steadily drift away from 50%.
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I'd think the proportion would tend to remain at the level it has achieved and spread out above and below it. Like a bunch of waves interfering with each other. The (1,0,1...) wave interferes with the (1,0,0...) wave. And so on.
There's got to be some kind of breakthrough insight that lets you simplify this. The N-tuples of 0,1's are no longer equally likely. But you might make an inductive argument according to the probability for classes C(N,n). ie. the class of tuples of length N containing n 1's. See how that probability progresses for C(3,n), C(4,n), ... deduce a closed form for it, then prove it works by induction.
PairTheBoard
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