Answer to Question #277951 in Statistics and Probability for michael

Question

A certain firm uses a large fleet of delivery vehicles. Their records over a long period of

time (during which their fleet size utilization may be assumed to have remained suitably

constant) show that the average number of vehicles per day is 3. Estimate the probability

on a given day when

i. all their vehicles will be serviceable (2 Marks)

ii. more than 2 vehicles will be unserviceable (2 Marks)

iii. exactly 4 vehicles will be unserviceable

Answer 1

Solution:

Let X = random variable denoting number of vehicles unserviceable per day.

$X\sim Poi(\lambda) \\ \lambda=3$

(i):

Probability that all their vehicles will be serviceable = 1 - Probability that all vehicles will be unserviceable

$=1-P(X=0) \\=1-\dfrac{e^{-3}(3)^0}{0!} \\=1-0.049787 \\=0.95021$

(ii):

Probability that more than 2 vehicles will be unserviceable

$=P(X>2)=1-P(X\le2) \\=1-[P(X=0)+P(X=1)+P(X=2)] \\=1-[\dfrac{e^{-3}(3)^0}{0!}+\dfrac{e^{-3}(3)^1}{1!}+\dfrac{e^{-3}(3)^2}{2!}] \\=0.57681$

(iii):

Probability that exactly 4 vehicles will be unserviceable

$=P(X=4) \\=\dfrac{e^{-3}(3)^4}{4!} \\=0.16803$