Question

The average number of motorcycles which stop at a petrol station is 5.2 per hour. By assuming that the number of motorcycles that stop at the petrol station follows a Poisson distribution, find the probability that

(a) 10 motorcycles stop at the petrol station at an interval of 60 minutes.

(b) more than 3 motorcycles stop at the petrol station at an interval of 10 minutes.

Answer 1

Mean λ=5.2

$P(X=k) = \frac{λ^k \times e^{-λ}}{k!}$

(a)

$P(X=10)= \frac{5.2^{10} \times e^{-5.2}}{10!} = 0.0219$

(b)

$Mean = \frac{5.2}{6}= 0.866 \\ P(X>3) = 1-P(X≤3) \\ = 1 -[P(X=0) +P(X=1) +P(X=2) +P(X=3)] \\ =1-[ (\frac{0.866^0 \times e^{-0.866}}{0!}) + (\frac{0.866^1 \times e^{-0.866}}{1!}) +(\frac{0.866^2 \times e^{-0.866}}{2!}) + (\frac{0.866^3 \times e^{-0.866}}{3!}) ] \\ = 1 -[e^{-0.866}(1+0.866+0.3749 + 0.1082)] \\ = 1-[0.4206 \times 2.3491] \\ = 1 -0.9880 \\ = 0.0012 \\ P(X>3) = 0.0012$