medium Probability
A release-checked medium problem for training Bayes' Theorem, Markov Chains.
A gambler starts with $\$k$ in a casino, where $k$ is a positive integer. The gambler bets $\$1$ per round in a fair game until they either have $\$m$, where $m \geq k$, or $\$0$. Given that the gambler ends with $\$m$, find the probability that they won the first round when $k = 10$ and $m = 20$. If necessary, round to the nearest thousandth.
This Probability problem is tagged Bayes' Theorem, Markov Chains. 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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