The fourth meteor past the half hour

Question

Meteors appear as a Poisson process at rate \lambda = 6 per hour. What is the probability that the 4th meteor of the night arrives more than 30 minutes after you start watching?

Accepted Answer

The 4th meteor arriving after 30 minutes is exactly the event of seeing fewer than 4 meteors (that is, 0, 1, 2, or 3) in the first half hour: P(	ext{4th arrival} 0.5) = P(N(0.5) \le 3). The count N(0.5) is Poisson with mean \mu = \lambda t = 6 	imes 0.5 = 3. So P(N \le 3) = e^{-3}\left(1 + 3 + \frac{3^2}{2!} + \frac{3^3}{3!}ight) = e^{-3}(1 + 3 + 4.5 + 4.5) = 13\,e^{-3} \approx 0.647. About a 65\% chance, so waiting past half an hour for the fourth meteor is actually the likely outcome here. Takeaway. The count trick handles any k: sum the Poisson probabilities from 0 up to k-1 at mean \l…

The fourth meteor past the half hour

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