I'm stuck on how to solve for a probability in this question!

Question

I've had problems with probability; I've been given this question on some homework of mine and I'm not quite sure how to even approach this.

"The average number of howler monkeys seen on a one-day excursion is 8. During a one-day excursion, find the probability that the number of howler monkeys seen is:

A) ten

B) between 5 and 11, inclusive"

I assume the average number of monkeys seen (the mean) is 8, but that's where my understanding stops completely. I don't know what formula to use here or even how to approach it.

Answer 1

This is a Poisson Distribution: mean = 8

X ~ Po(lambda = 8)

a) Thus, P(X = 10) = e^-8 (8¹⁰/10!)

b) P (5 < X < 11) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)

[then same things for (a)]

Answer 2

A Poisson distribution models the number of events occurring in a fixed interval of time when the events are independent and the average rate of the events is known.

Conducting a Poisson experiment, in which the average number of successes in a given region is mu, then the Poisson probability is:

P(x;mu) = e **-mu * mu**x/x!

where x is the actual number of successes that result from the experiment (e=2.71828).

For part a) P(10,8) = e**-8 * 8**10/10!

Or you can use the Poisson Calculator at https://stattrek.com/online-calculator/poisson.aspx and enter 10 for Poisson variable and 8 for average rate of successes and get P(x=10) = 0.09926.

For part b) we would need to use the above calculator and compute P(<= 11, 8) - P(<=5,8) = 0.88808 - 0.19124 = 0.69684.