Chapter PIV, Problem 40RE

(a)

To determine

To find: the probability that a arbitrary selected car is recalled.

Answer to Problem 40RE

0.017

Explanation of Solution

Given:

  $\begin{array}{c}P\left(A\right)=0.7\\ P\left(J\right)=0.2\\ P\left(G\right)=0.1\\ P\left(R|A\right)=0.02\\ P\left(R|J\right)=0.01\\ P\left(R|G\right)=0.01\end{array}$

Calculation:

Suppose

A = car is American brand

J = car is Japanese brand

G = car is German brand

R = car is recalled

It is given that

  $\begin{array}{c}P\left(R\right)=P\left(A\right)P\left(R|A\right)+P\left(J\right)P\left(R|J\right)+P\left(G\right)P\left(R|G\right)\\ =\left(0.7\right)\left(0.02\right)+\left(0.2\right)\left(0.01\right)+\left(0.1\right)\left(0.01\right)\\ =0.017\end{array}$

(b)

To determine

To find: the probability of a recalled car being American.

Answer to Problem 40RE

0.82

Explanation of Solution

0.82

Given:

  $\begin{array}{c}P\left(A\right)=0.7\\ P\left(J\right)=0.2\\ P\left(G\right)=0.1\\ P\left(R|A\right)=0.02\\ P\left(R|J\right)=0.01\\ P\left(R|G\right)=0.01\end{array}$

Calculation:

  $\begin{array}{c}P\left(A|R\right)=\frac{P\left(A\cap R\right)}{P\left(R\right)}\\ =\frac{P\left(A\right)P\left(R|A\right)}{P\left(R\right)}\\ =\frac{\left(0.7\right)\left(0.02\right)}{0.017}\\ =0.82\end{array}$