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Old 2018-04-04, 22:07   #5
wblipp's Avatar
May 2003
New Haven

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Originally Posted by Nick View Post
How many ways are there of throwing \(k-N\) times without getting \(N\) heads in a row? What is the probability of each of those possibilities followed by \(N\) consecutive heads?
Not quite right, either. The \(k-N\) might have ended with one or more heads. I usually grab the density using the transition matrix for the Markov Chain of #heads in a row with an absorbing state at \(N\), but this question asks for the mean.

I'd try an inductive approach on the number of heads. To get one head in a row, the first toss is a head with probability p or a tail with probability (1-p). If heads, you are done. If tails, you need the to start over. So

X1 = p*1 + (1-p)*(X1+1)
X1 = 1/p

For k in a row, you need k-1 in a row, then one more toss either finishes it or starts it over.

Xk = p*(X(k-1)+1) + (1-p)(Xk+X(k-1)+1)
Xk = X(k-1)/p + 1/p

The logic of the last equation is conditioned on the results of the flip after first achieving k-1 heads.

Check that arithmetic - I fixed several errors before posting and might still have some left - but the logic is solid.
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