Question

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.]

Answer 1

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: $P(X \ge 3) \approx 0.3233$