Three events in two hours

Topic: Poisson processes · more questions on this topic
‹ Counts and a first wait Distance to the nearest point ›

Given N(2)=3, find the probability that all three events occurred in the first hour. State the conditioning.

Worked answer (click to show)

Conditional on N(2)=3, the three unordered event times are independent and uniform on (0,2). Each falls in the first hour with probability 1/2, independently, so the probability that all three do is (1/2)^3=1/8.