Q.4.54 - A First Course in Probability

Question

Suppose that the average number of cars abandoned weekly on a certain highway is $2.2$. Approximate the probability that there will be

(a) no abandoned cars in the next week;

(b) at least abandoned cars in the next week.

Step-by-Step Solution

Answer

The probability that there will be no abandoned cars in the next week is$0.1108.$

The probability that there will be at least $2$ abandoned cars in the next week is$0.645.$

Step 1: Given Information (Part-a)

Given in the question that the average number of cars abandoned weekly on a certain highway is$2.2$. The probability that there will be no abandoned cars in the next week.

Step 2: Solution of the Problem (Part-a)

Given $\lambda =2.2$

$\mathrm{P}(\mathrm{X}=0)=\frac{{\mathrm{e}}^{-\lambda }(\lambda {)}^0}{0!}$

We get,

$=\frac{e^{-2.2}2.2^0}{0!}$

$=0.1108.$

Step 3: Final Answer (Part-a)

The probability that there will be no abandoned cars in the next week is $0.1108.$$0.1108.$

Step 4: Given Information (Part-b)

Given in the question that the average number of cars abandoned weekly on a certain highway is $2.2.$The probability that there will be at least $2$abandoned cars in the next week.

Step 5: Solution of the Problem (Part-b)

Given $\lambda =2.2$

$P(X\ge 2)=1-P(X\le 1)$

$=1-\left[\mathrm{P}\right(\mathrm{X}=0)+P(X=1\left)\right]$

We get,

$=1-[0.1108+0.2438]$

$=0.645$.

Step 6: Final Answer (Part-b)

The probability that there will be at least $2$ abandoned cars in the next week is $0.645.$