Did the first of two customers come early?

Question

Customers enter a boutique as a Poisson process over a $30$ -minute window, $[0, 30]$ minutes. Exactly $2$ customers came in.

Given this, what is the probability that the first customer arrived within the first $5$ minutes?

Accepted Answer

Given 2 customers, their times act like 2 independent uniform points on [0, 30]. The "first customer" is the earlier of the two, and we want that earliest time to be under 5 minutes. It is cleanest to use the complement. The first customer arrives after 5 minutes only if both customers arrive after 5 minutes. Each independently lands in the later 25-minute stretch with probability 	frac{25}{30} = 	frac56, so P(	ext{both after 5 min}) = \left(\frac56ight)^2 = \frac{25}{36}. Therefore P(	ext{first within 5 min}) = 1 - \frac{25}{36} = \frac{11}{36} \approx 0.306. About a 31\% chance. Takea…

Did the first of two customers come early?

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