Question

4. The number of spectators that turn up to support their local cricket team is known to have a Poisson distribution with mean 10. The admission charge is $3.00 per person, and the gateperson is paid $16.00 for collecting the admission money. (a) Determine the probability that the team makes a profit [3] (Write your answer in the box provided and show any working below.)

Answer 1

Let X represent the number of people who turn up. Therefore the distribution for X here is given as:

XPoisson(10)

Now the profit is computed as:

= Total Revenue - Total Cost

= 3X - 16

because 16 is the total cost that is the gatekeeper's fees.

a) Probability that there is a profit is computed as:

P( 3X > 16)

= P(X > 16/3)

= 1 - P(X <= 5)

This is computed using the poisson probability function as:

102 103 104 105 -10 -10 -10 e 2 -10 e 3! -10 e -10 e 1 -6 10e - 4! 5!

0.93291

Therefore 0.93291 is the required probability here.

b) The expected value of the profit is computed as the expected value of the random variable P as:

E(P) = E(3X - 16) = 3E(P) - 16 = 3*10 - 16 = 14

Therefore $14 is the expected profit here.

c) The variance of the profit here is computed as:

Var(P) = Var(3X - 16) = 3² Var(X) = 9*10 = 90

Note that the variance for a poisson random variable is same as the mean of that variable.