A traffic engineer records the number of bicycle riders that use a particular cycle track. He records that an average of 3.2 bicycle riders use the cycle track every hour. Given that the number of bicycles that use the cycle track follow a Poisson distribution, what is the probability that 3 or more bicycle riders will use the cycle track within an hour?

0.618

The correct answer is Option (3) → 0.618

Given mean = $λ = 3.2$

Let X be the number of bicycle riders which use the cycle track.

$P(X ≤2) = P(X = 0) + P(X = 1) + P(X = 2)$

$=\frac{e^{-3.2}(3.2)^0}{0}+\frac{e^{-3.2}(3.2)^1}{1}+\frac{e^{-3.2}(3.2)^2}{2}$

$=e^{-3.2} [1 +3.2 + 5.12]$

$=0.041 × 9.32 = 0.382$

Required probability = $P(X ≥ 3) = 1-P(X ≤2)$

$= 1-0.382 = 0.618$