Question

In a certain area an average of 13 new swarms of honeybees are seen each spring. If the number of swarms stays constant each year, what is the probability of observing between 9 and 15 (inclusive) swarms?

Answer 1

The probability of observing between 9 and 15 (inclusive) swarms is 0.6639.

Explanation:

The random variable X can be defined as the number of swarms of honeybees seen each spring.

The average value of the random variable X is, λ = 13.

A random variable representing the occurrence of events in a fixed interval of time is known as Poisson random variables.

For example, the number of customers visiting the bank in an hour or the number of typographical error is a book every 10 pages.

So, the random variable X follows a Poisson distribution with parameter λ = 13.

The probability mass function of X is as follows:

`P(X=x)=(e^(-\lambda)\ \lambda^(x))/(x!); x=0,1,2,3...`

Compute the the probability of observing between 9 and 15 (inclusive) swarms as follows:

P (9 ≤ X ≤ 15) = P (X = 9) + P (X = 10) + P (X = 11) + ... + P (X = 15)

`=\sum\limits^(15)_(x=9){(e^(-\lambda)\ \lambda^(x))/(x!)}\\\\=0.06605+0.08587+0.10148+0.10994\\+0.10994+0.10209+0.08848\\\\=0.66385\\\\\approx 0.6639`

Thus, the probability of observing between 9 and 15 (inclusive) swarms is 0.6639.