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Bob is to meet Joe between 11:30 am and noon. If they arrive at random times during this interval and their arrivals are independent, what's the probability that they'll have to wait for each other at most 10 minutes?

I understand that the events of Bob and Joe arriving are Poisson processes, with $\lambda = 1/30$ in both cases, and that the time between Bob's arrival and Joe's arrival can be described by an random variable with an exponential distribution. How do I set up this problem?

David G. Stork
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blargen
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2 Answers2

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Try plotting out the 30 minute period on two axes; graphically, what's the region of that square that is a "success"? What's the probability that a point from that square is in the success region?

khu
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  • We can split up the region of success into three regions. If we let B be the time that Bob arrives and J be the time that Joe arrives in minutes after 11:30am, we can describe the first region by 0<= B <= 10, 0<= J<= B+10; the second by 10<=B<=20, B-10<=J<= B+10; and the third by 20<= B <= 30, B-10<= J <= 30. – blargen Jun 21 '17 at 22:37
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It would be a mistake to try to model this as a Poisson process because such a process would allow 0, 1, 2, ... "arrivals" of Bob and Joe. As I read your problem, you posit that exactly one of each will arrive in the allotted half-hour interval--no more, no fewer.

As described by Kevin Hu, each point in the arrival space is equally likely, and the "success" region is in blue:

enter image description here

The relative "success" area--and hence probability--is $5/9$.

David G. Stork
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