Earthquakes occur in the western United States with a rate of 2 per week. If we model the number of earthquakes as a Poisson random variable, what is the probability that there will be at least 3 earthquakes in a week period? [correct your answer to 4 d.p.]

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By condition “\lambda = 2”.

Let’s find the probability that there will be less than three earthquakes (0, 1 or 2).

“P(0) = \frac{{{\lambda ^0}}}{{0!}}{e^{ – \lambda }} = {e^{ – 2}} = \frac{1}{{{e^2}}}”

“P(1) = \frac{{{\lambda ^1}}}{{1!}}{e^{ – \lambda }} = \frac{2}{{{e^2}}}”

“P(2) = \frac{{{\lambda ^2}}}{{2!}}{e^{ – \lambda }} = \frac{2}{{{e^2}}}”

Then

“P(X < 3) = P(0) + P(1) + P(2) = \frac{{1 + 2 + 2}}{{{e^2}}} = \frac{5}{{{e^2}}}”

Then the wanted probability, as the probability of the opposite event, is

“P(X \ge 3) = 1 – P(X < 3) = 1 – \frac{5}{{{e^2}}} \approx 0.3233”

Answer: “P(X \ge 3) \approx 0.3233”

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