26.The Bus That Stops for Ice Cream
★★★expected valueJane Street
A bus runs a circuit that normally takes 10 minutes. Each time it passes an ice cream shop on the route the driver tosses a fair coin, and on heads spends an extra 10 minutes eating ice cream before carrying on. So each loop takes 10 or 20 minutes with equal probability, independently. You arrive at the stop at a uniformly random moment in time. What is your expected waiting time in minutes?
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You are more likely to land inside a long gap than a short one, simply because long gaps occupy more of the timeline. Weight the cycles by their length, not by their count.
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