How many times on average (expected value) must we flip $6$ fair coins before we obtain $3$ heads and $3$ tails?
I know I need $∑ xp(x)$. I just don't know how to apply it.
2026-04-07 09:26:56.1775554016
Expected value problem: flip $6$ fair coins before we obtain $3$ heads and $3$ tails?
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Call three heads and three tails when tossing six coins a success. Then the probability of success is $\frac{\binom{6}{3}}{2^6}$, which simplifies to $\frac{5}{16}$.
Let $X$ be the number of trials until the first success. Then $X$ has geometric distribution with parameter $p=\frac{5}{16}$.
It is a standard result that if $X$ has geometric distribution with parameter $p$, then $E(X)=\frac{1}{p}$.