Chapter 3.2, Problem 13E

a.

To determine

Delineate whether $P\left(AandB\right)$ can be obtained if $P\left(A\right)\text{ and }P\left(B\right)$ are known.

Answer to Problem 13E

No, it can be obtained only if $A\text{ and }B$ are independent.

Explanation of Solution

From the given information, $P\left(A\right)=0.3,P\left(B\right)=0.7$.

General multiplication rule:

If A and B are two events, then

$P\left(\text{A and B}\right)=P\left(A|B\right)\times P\left(B\right)$

Using the given information , $P\left(AandB\right)$ cannot be obtained because the value of $P\left(A|B\right)$ is not known.

If A and B are independent events, then $P\left(A\text{ and }B\right)=P\left(A\right)\times P\left(B\right)$.

Thus, $P\left(AandB\right)$ can be obtained only if A and B are independent.

b.

To determine

If A and B are independent events, then find $P\left(AandB\right).$

If A and B are independent events, then find $P\left(AorB\right).$

If A and B are independent events, then find $P\left(A|B\right)$.

Answer to Problem 13E

$\begin{array}{l}P\left(A\text{ and }B\right)=\underset{\_}{0.21}\\ P\left(A\text{ or }B\right)=\underset{\_}{0.79}\\ P\left(A|B\right)=\underset{\_}{0.3}\end{array}$

Explanation of Solution

From the given information, $A\text{ and }B$ are independent events. $P\left(A\right)=0.3,P\left(B\right)=0.7$

Assuming $A\text{ and }B$ are independent events, then using multiplication rule for independent process:

$\begin{array}{c}P\left(AandB\right)=P\left(A\right)\times P\left(B\right)\\ =0.3\times 0.7\\ =0.21\end{array}$

Thus, if $A\text{ and }B$ are independent events, then $P\left(AandB\right)=0.21$.

Assuming $A\text{ and }B$ are independent events, then using the general addition rule:

$\begin{array}{c}P\left(A\text{or}B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)\\ =0.3+0.7-P\left(A\right)\times P\left(B\right)\\ =1-\left(0.3\times 0.7\right)\\ =1-0.21\\ =0.79\end{array}$

Thus, if $A\text{ and }B$ are independent events, then $P\left(A\text{or}B\right)=0.79$.

Assuming $A\text{ and }B$ are independent events, then using conditional probability:

$\begin{array}{c}P\left(A|B\right)=\frac{P\left(A\text{ and }B\right)}{P\left(B\right)}\\ =\frac{P\left(A\right)\times P\left(B\right)}{P\left(B\right)}\\ =P\left(A\right)\\ =0.3\end{array}$

Thus, if $A\text{ and }B$ are independent events, then $P\left(A|B\right)=0.3$.

c.

To determine

Check whether the random variables A and B are independent if $P\left(A\text{ and }B\right)=0.1$.

Answer to Problem 13E

A and B are not independent events.

Explanation of Solution

Given information, $P\left(A\right)=0.3,P\left(B\right)=0.7\text{ and }P\left(A\text{ and }B\right)=0.1$.

Multiplication rule for independent process:

$\begin{array}{c}P\left(A\text{ and }B\right)=P\left(A\right)\times P\left(B\right)\\ =0.3\times 0.7\\ =0.21\end{array}$

Given that $P\left(A\text{ and }B\right)=0.1$.

Here,

$\begin{array}{l}0.1\ne 0.21\\ P\left(A\text{ and B}\right)=P\left(A\right)\times P\left(B\right)\end{array}$

Thus, A and B are not independent events.

d.

To determine

Find $P\left(A|B\right)$.

Answer to Problem 13E

$P\left(A|B\right)=0.143$

Explanation of Solution

Given information, $P\left(A\right)=0.3,P\left(B\right)=0.7\text{ and }P\left(A\text{ and }B\right)=0.1$.

Conditional probability:

$\begin{array}{c}P\left(A|B\right)=\frac{P\left(A\text{ and }B\right)}{P\left(B\right)}\\ =\frac{0.1}{0.7}\\ =0.14285\\ \approx 0.143\end{array}$

Thus, $P\left(A|B\right)=0.143$