Binomial and Poisson distribution

Question

Suppose that N tickets are sold for a particular live music event but that there are only M seats available (M < N) with standing not allowed. Assume that each person does not show up, independently of the others, with probability 0.05. Evaluate (to 4 decimal places) the exact Binomial probability and the corresponding Poisson approximation probability that there will be enough seats for all the people who show up based on the following values of N and M:

(a) N = 50, M = 48

(b) N = 500, M = 480

Answer 1

(a) N~B(50, 0.05)

So P(X<=48) N=50, p=0.05

P(X<=48) = 1

and

N~P(2.5)

So P(X<=48) λ= 50*0.05= 2.5

P(X<=48)= 1

(b) N~B(500, 0.05)

So P(X<=480) N=500, p=0.05

P(X<=480) = 1

and

N~P(25)

So P(X<=480) λ= 500*0.05= 25

P(X<=480) = 1