Help me solve this question

Question

Post boxes in London are randomly spaced out and on average five post

boxes are found per square mile.

(a) Determine the probability that on a certain square mile there will be

(i) exactly seven post boxes

(ii) less than eight but no less than four post boxes.

The number of post boxes in a small area of ¼

of a square mile are counted.

(b) Find the probability that this smaller area will contain post boxes.

Answer 1

The is a Poisson distribution problem: P(X=x) = e^-µµ^x/x! where µ is the mean.

Here µ = 5 post boxes per square mile.

a)

i) P(X=7) = e^-55⁷/7! = 0.1044

ii) P(4≤X<8) = P(X=4) + P(X=5) + P(X=6) + P(X=7) = 0.17547 + 0.17547 + 0.14622 + 0.10444

= 0.6016

b) Now µ = 5/4 = 1.25. To contain post boxes mean to have at least 1 post box, i.e. P(X≥1) is being asked.

P(X≥1) = 1 - P(X=0) = 1 - e^-1.251.25⁰/0! = 1 - e^-1.25 = 0.7135