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Find the expected value of the number of coin flips needed to get two consecutive heads.

So here we're dealing with a geometric random variable. I know that the expectation of such a variable is $1/p$, where $p$ is the probability of success. So it suffices to find $p$, but I'm not sure how to do this.

To begin with, does "success" in this case correspond to getting two heads in a row? If so, how to compute it? If we had a small finite number of coin flips, then the probability of two consecutive heads could be easily found from a tree. But in this case we have potentially infinite number of coin flips, so the tree is infinite, and I don't know how to proceed.

user125234
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  • There are many, many variants of this question on this site. [here](https://math.stackexchange.com/questions/364038/expected-number-of-coin-tosses-to-get-five-consecutive-heads/364135#364135) for instance. or [here](https://math.stackexchange.com/questions/1155104/expected-number-of-coin-tosses-to-land-n-heads). – lulu Feb 24 '23 at 00:06
  • @lulu But as far as I can see, the answers in those questions don't answer how to calculate my $p$, they only explain how to calculate the expectation by other means. – user125234 Feb 24 '23 at 03:03
  • I don't understand the method you propose, it does not seem relevant. This is not a geometric process, that requires independent trials. We haven't got that here, not in the sense that matters. Success on a given trial requires a specific result from the prior trial. – lulu Feb 24 '23 at 11:44

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