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Total Expectation: Poisson and Negative Binomial

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A number N is chosen according to a Poisson distribution with mean 10. A fair coin is then tossed until N +1 heads are obtained. What is the expected number of tosses it will take to stop the experiment?

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A full step-by-step explained solution is given, with references to websites for more detail.

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What you are given is the following information:

N ~ Poisson(10)
X | N = n ~ negative binomial(n+1, 0.5)

The second expression is basically saying that, conditional on knowing N = n, X follows a negative binomial distribution ...

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