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Henry and Ann are waiting for a bus. They know from experience that if they wait for an hour, they will have a 90% chance of getting it. It is a chilly night, though, so Ann says, "Let's only stay out for 10 minutes."

Henry says, "If we only wait for 10 minutes, we will only have a 15% chance."

Ann replies, "Not true. We have a better chance than that." Is Ann right? If so, what is the probability that they getting on the bus?

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If you assume the buses arrive regularly and Henry and Ann arrive at a random moment in the cycle (very strong assumptions, but we can do no better), the buses must come every $\frac {10}9$ hour. If they wait $10$ min $=\frac 16$ hour, they have $\dfrac {\frac 16}{\frac {10}9}=\frac 9{60}=0.15$ chance of getting a bus, so Henry is right. They might do well to study the bus schedule.

If the buses arrive randomly, the rate is to get one with $90\%$ chance in one hour is $\lambda=\ln 10$. The chance for one not to arrive in $\frac 16$ hour is $e^{\frac{-\lambda}6}\approx 0.68$ so Ann would be right-they would have $32\%$ chance of getting one.

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Or rather to study the actual distribution of buses, which may or may not be related to the schedule. – Chris Eagle Feb 1 '13 at 16:34
Hold on, if the buses arrive randomly, isn't the rate $\lambda = \ln(10)$ per hour? – Erick Wong Feb 1 '13 at 16:37
@ErickWong: Right you are. Fixed. – Ross Millikan Feb 1 '13 at 16:52

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