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The problem I'm currently looking over requires use of generating functions to solve the following:

If a coin is flipped $25$ times with eight tails occurring, what is the probability that no run of six (or more) consecutive heads occurs?

I know that you need to designate $H$ (heads) and $T$ (tails), such as $H = 1$, $T = x$ and this would give you the binomial expression $(1+x)^n$, but I'm not entirely sure how to handle the constraint of tails already occurring and then find a probability of six (or more) consecutive heads from that.

I appreciate the help!

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Hint: the 8 tails partition the heads into 9 (possibly empty) runs.

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Thanks, i'll take a look using partitions – CSDUDE Apr 24 '13 at 15:36

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