Question

Suppose that the number of weekly traffic accidents occurring in a small town is Poisson random variable with parameter lamda = 7.

(a) what is the probability that at leat 4 accidents occur (until) this week?

(b) what is the probability that at most 5 accidents occur (until) this week given that at least 1 accident will occur today.

Answer 1

Solution :

Let X represents the number of weekly traffic accidents occuring  in a small town.

Given that, X ~ Poisson (7)

According to poisson distribution, probability of occurrence of exactly x events is,

a) We have to obtain P(at least 4).

We have, lambda = 7

P(at least 4) = P(X ≥ 4)

P(X ≥ 4) = 1 - P(X < 4)

P(X ≥ 4) = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)]

Using poisson probability law we get,

The probability that at leat 4 accidents occur (until) this week is 0.9182.

b) We have to obtain P(X = at most 5 | X ≥ 1).

P(X = at most 5 | X ≥ 1) = P(X ≤ 5 | X ≥ 1)

Now,

P(X ≥ 1) = 1 - P(X < 1)

P(X ≥ 1) = 1 - P(X = 0)

The probability that at most 5 accidents occur (until) this week given that at least 1 accident will occur today is 0.3001.