Taco truck or burger truck first?

Question

A taco truck draws customers as a Poisson process at rate $8$ per hour, and a nearby burger truck draws customers independently at rate $12$ per hour.

What is the probability that the very next customer (at either truck) shows up at the taco truck?

Accepted Answer

Think of two independent Poisson processes racing to produce the next event. A neat fact: the probability that a given process wins the race is just its share of the total rate. The intuition is that the combined stream arrives at rate 8 + 12 = 20 per hour, and each arrival is independently assigned to a truck in proportion to that truck's rate. So P(	ext{taco first}) = \frac{\lambda{	ext{taco}}}{\lambda{	ext{taco}} + \lambda{	ext{burger}}} = \frac{8}{8 + 12} = \frac{8}{20} = 0.4. A 40\% chance. Notice the answer depends only on the ratio of rates, not on how much time passes or how busy …

Taco truck or burger truck first?

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