Law of total variance

MediumConditional Probability & Expectation Expected Value Statistics~6m

A coin's bias $P$ is uniform on $[0,1]$ ; given $P=p$ you flip once, scoring $X=1$ for heads. Using the law of total variance, what is $\mathrm{Var}(X)$ ? (to 2 decimals)

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Var(X) = E[Var(X|P)] + Var(E[X|P]).
E[P(1-P)] = 1/2 - 1/3 and Var(P) = 1/12.

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0.25 (± 0.01)

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Law of total variance

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