Which alarm rings next?

Question

Two independent alarms fire as Poisson processes: alarm A at rate $2$ per hour and alarm B at rate $3$ per hour.

When the next beep sounds (from either alarm), what is the probability it came from alarm A?

Accepted Answer

Merge the two streams. Independent Poisson processes combine into a single Poisson process whose rate is the sum: \lambda{	ext{total}} = 2 + 3 = 5 	ext{ per hour}. Each event in the merged stream independently "belongs" to a source in proportion to that source's rate. So the probability the next beep is from alarm A is simply A's share of the total rate: P(	ext{next is A}) = \frac{\lambdaA}{\lambdaA + \lambdaB} = \frac{2}{2 + 3} = \frac{2}{5} = 0.4. Equivalently, the wait for A's next beep is Exponential(2) and for B's is Exponential(3); the probability A's comes first is \frac{2}{2+3}, th…

Which alarm rings next?

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