Consider the experiment of keeping flipping a fair coin until two heads are flipped consecutively. The outcome is the sequence of the flipped result. Possible examples of the outcome include HTHTHH and TTTHTHTTHH. The random variable X is defined to be the number of flips of an outcome. For example, and X(HTHTHH)=6 and X(TTTHTHTTHH)=10. Find E(X).
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Let $x$ be the number of expected flips to get heads.
If you flip a tails, you have made one flip, and you still expect to flip x more times.
If you flip heads, you are close to being done. But if you flip HT you are back to square one and x flips away.
If you flip HH you are done.
$x = \frac 12 (x+1) + \frac 14 (x+2) + \frac 14 (2)$
Doug M
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Ummm... what about HTHTHTHTHTH? – David G. Stork Nov 07 '19 at 06:27
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@DavidG.Stork for every long one like that you have a short one like THH to balance out the average to 6. – Doug M Nov 07 '19 at 07:27
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But "balancing out" means they must have precisely the same probability of occurring. – David G. Stork Nov 07 '19 at 14:51