[pzhon]

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To me, it seems as though there cannot be a single correct answer because you cannot directly relate the median and the mean. It seems to me that you would need to make additional assumptions.

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Come up with a reasonable model of waiting times with one parameter. Analyze your model. Specifically, compute the conditional expectation of the waiting time given that you are still waiting after 5 minutes.

One model might be that the waiting times follow an exponential distribution. That's simple because it is memoryless, so you can ignore the initial wait. With a half-life of 15 minutes, the expected wait is not 15 minutes, but is easily computed. I think that is the answer the professor wants. However, if this were a more serious problem, you might want to use a better model (perhaps based on a queue), and you would get a different answer.