Which elevator arrives first?

Question

In a lobby, the left elevator arrives at your floor as a Poisson process at rate $3$ per hour, and the right elevator independently at rate $9$ per hour.

What is the probability that the next elevator to arrive is the left one?

Accepted Answer

The two elevators are independent Poisson processes racing to arrive next. Each wins in proportion to its own rate: P(	ext{left first}) = \frac{\lambda{	ext{left}}}{\lambda{	ext{left}} + \lambda{	ext{right}}} = \frac{3}{3 + 9} = \frac{3}{12} = 0.25. A 25\% chance. The right elevator is three times as frequent, so it wins three times as often, giving the left just one in four. Takeaway. In a race between Poisson processes, win-probability is the rate share; a process three times as fast wins three times as often.

Which elevator arrives first?

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