medium Probability
A release-checked medium problem for training Stochastic Processes, Conditional Expectation.
A frog has $100$ lily pads arranged in a circle. Each one has a distinct number $0,1,\dots, 99$ on it, arranged in increasing order counter-clockwise starting from $0$. The frog starts at lily pad $0$ and hops to the closest lily pad on the left or right of its current position with equal probability per hop. Find the probability that when the frog lands on lily pad $50$ for the first time, it has visited every other lily pad not labeled $50$.
This Probability problem is tagged Stochastic Processes, Conditional Expectation. 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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