Find the variance for the number of boys and the number of girls in a royal family that has children until there is a boy or until there are three children, whichever comes first.

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

Solution: C

Let [math]N[/math] be the number of boys. Clearly [math]N \leq 1 [/math]. If [math]N=0 [/math] then we have three girls and the probability of this event is (1/2)³. Hence the probability distribution for [math]N[/math] is [math]P(N=0) = 0.125, P(N=1) = 0.875 [/math]. Then [math]E[N] = 0.875 [/math] and [math]E[N^2] = E[N] = 0.875 [/math] and the variance equals 0.875 - 0.875² = 0.109375.