medium Probability
A release-checked medium problem for training Coins, Uniform, Expected Value/LOTUS, Geometric/Negative Binomial.
A family births children under the following scheme: They birth children one-at-a-time until the first boy is born, and then they stop. Assuming each child is a boy with probability $0 < p < 1$, find the expected proportion of the family's children that are boys when $p = 2/3$. Round to the nearest thousandth.
This Probability problem is tagged Coins, Uniform, Expected Value/LOTUS, Geometric/Negative Binomial. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.
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