medium Probability
A release-checked medium problem for training Conditional Expectation, Law of Large Numbers, Expected Value/LOTUS, Conditional Probability.
Alice and Bob play in a tennis match indefinitely. One of them starts serving at random, and the winner of the previous point is who serves in the next point. Alice wins each point she serves with probability $p_1$, while she wins every point Bob serves with probability $p_2$. Find the proportion of points that are won by Alice. Report the answer when $p_1 = 2/3$ and $p_2 = 1/3$.
This Probability problem is tagged Conditional Expectation, Law of Large Numbers, Expected Value/LOTUS, Conditional Probability. 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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