medium Probability
A release-checked medium problem for training Law of Total Probability, Expected Value/LOTUS, Conditional Probability, Recurrence Relations.
George starts with $\$1$. George repeatedly plays a game where he has probability $0 < p < 1$ of winning, independently between rounds. If George wins, he receives $\$r$. If George loses, his current balance is multiplied by $0 < q < 1$. If George plays infinitely, what does his expected balance converge to? Report the answer when $p = 1/3, q = 1/2,$ and $r = 3$.
This Probability problem is tagged Law of Total Probability, Expected Value/LOTUS, Conditional Probability, Recurrence Relations. 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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