A lightning strike within five minutes

Question

During a storm, lightning strikes near a tall tower as a Poisson process at rate \lambda = 6 per hour. What is the probability that the next strike happens within the next 5 minutes?

Accepted Answer

This time we want the wait to be short rather than long. The chance the wait exceeds t is e^{-\lambda t}, so the chance it happens within t is the complement: P(T \le t) = 1 - e^{-\lambda t}. Five minutes is 	frac{5}{60} = 	frac{1}{12} of an hour, and \lambda = 6 per hour, so \lambda t = 6 	imes 	frac{1}{12} = 0.5 and P(T \le 	frac{1}{12}) = 1 - e^{-0.5} \approx 1 - 0.6065 = 0.393. About a 39\% chance of a strike in the next five minutes. Takeaway. For "happens within t," take one minus the exponential tail: P(T \le t) = 1 - e^{-\lambda t}.

A lightning strike within five minutes

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