Solution
Two Buses in a Row
Show the problem again
Three bus lines stop at a corner. Buses of line A arrive as a Poisson process with rate 1 per hour, line B with rate 2 per hour, and line C with rate 3 per hour, all independent. You arrive at a random moment. What is the probability that the next two buses to arrive are both line A? Give a reduced fraction.
Worked solution
The answer is 1/36. The superposition of the three processes is a Poisson process of rate 6, and each arrival is independently labelled A with probability 1/6 (competing exponentials: the first arrival is A with probability 1/(1+2+3)). By memorylessness everything restarts after the first bus, so the second bus is again A with probability 1/6, independently. Hence (1/6)² = 1/36.
Source: Classic exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.