Question

On average, 3 traffic accidents per month occur
at a certain intersection. What is the probability that
in any given month at this intersection
(a) exactly 5 accidents will occur?
(b) fewer than 3 accidents will occur?
(c) at least 2 accidents will occur?

Answer 1

Question #72702, Math / Statistics and Probability

Solution: - Here we have to make an assumption that number of accidents at a certain intersection follows Poisson distribution. Let $X$ be the random variable denoting the number of accidents in a month at a particular intersection.

So, here $X \sim$ Poisson (3) since, mean is given to be 3.

$$P(X = a) = \left(\frac{e^{-3} 3^a}{a!}\right)$$

We need to find probability of

a) Exactly 5 accidents: - $P(X = 5) = \left(\frac{e^{-3} 3^5}{5!}\right)$

b) Fewer than 3 accidents occur: -

$$\begin{array}{l} P(X = 0) + P(X = 1) = \frac{e^{-3}}{3} + 3 \frac{e^{-3}}{3} \\ = 4 \frac{e^{-3}}{3} \end{array}$$

c) At least two accidents occur: -

$$\begin{array}{l} P(X \geq 2) = 1 - P(X = 0) - P(X = 1) \\ = 1 - \frac{e^{-3}}{3} - 3 \frac{e^{-3}}{3} \\ = 1 - 4 \frac{e^{-3}}{3} \end{array}$$

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