A couple decide to start having children and keep having children until they have more girls than boys (this means they stop having children if their first child is a girl). How many children do they expect to have? I've tried the first few simple cases, and find that having one child has one probability(g),three has one(bgg),5 has 2(bbggg,bgbgg),7 has 5(bbbgggg,bbgbggg,bbggbgg,bgbbggg,bgbgbgg).and the formula is $1*1/2+3*(1/2)^3+5*(1/2)^5*2+7*(1/2)^7*5+etc$ but I couldn't find a pattern to solve it.
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This is closely related to the https://en.wikipedia.org/wiki/Bertrand%27s_ballot_theorem – b00n heT Oct 12 '23 at 05:58
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2See Expected number of steps to first reach K in a random walk. You want $K=1$. – Calvin Lin Oct 12 '23 at 06:15