Is the first particle type A?

Question

A detector receives type-A particles as a Poisson process at rate $6$ per second and type-B particles independently at rate $2$ per second.

What is the probability that the first particle detected is type A?

Accepted Answer

The two particle streams are independent Poisson processes racing to be detected first. Each type wins in proportion to its rate: P(	ext{A first}) = \frac{\lambdaA}{\lambdaA + \lambdaB} = \frac{6}{6 + 2} = \frac{6}{8} = 0.75. A 75\% chance. Type A arrives three times as fast as type B, so it is three times as likely to come first, which is exactly the 6\!:\!2 split. Takeaway. The faster process wins the race with the larger rate share; here 6\!:\!2 odds give type A a 75\% chance of being first.

Is the first particle type A?

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