A traffic police officer records the number of motorcycle riders that use a particular lane on a highway. He records that an average of 2.5 motorcycle riders use the lane every hour. Given that the number of motorcycles that use the particular lane follows a Poisson distribution. The probability that less than 2 motorcycle riders will use the lane within an hour will be:- (Use e2.5 = 12.2)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 35/122
For a Poisson distribution with mean λ, \( P(X=r)=\frac{e^{-\lambda}\lambda^r}{r!} \). Here λ = 2.5, and "less than 2" means X = 0 or X = 1.
\( \displaystyle P(X\lt 2)=P(X=0)+P(X=1) \)
\( \displaystyle =e^{-2.5}\left(1+2.5\right) \)
\( \displaystyle =\frac{3.5}{e^{2.5}} \)
\( \displaystyle =\frac{3.5}{12.2} \)
\( \displaystyle =\frac{35}{122} \)
Why the other options are incorrect
Option (1) 5/61 — Incorrect. \( \frac{5}{61}=\frac{10}{122}=\frac{1}{12.2} \) is only P(X = 0).
Option (2) 625/244 — Incorrect. \( \frac{625}{244} \) is greater than 1, so it cannot be a probability.
Option (4) 25/122 — Incorrect. \( \frac{25}{122}=\frac{2.5}{12.2} \) is only P(X = 1).