Chapter 4, Problem 12CRP

a.

To determine

Find the values of $P\left(N\right)$, $P\left(M\right)$ and $P\left(S\right)$.

Answer to Problem 12CRP

The value of $P\left(N\right)$ is 0.470.

The value of $P\left(M\right)$ is 0.390.

The value of $P\left(S\right)$ is 0.140.

Explanation of Solution

It is given that the events W, A, S, M and N are defined as follows:

$W=\text{washing oil off within 5 minute}$, $A=\text{washing oil off after 5 minute}$

$S=\text{Strong reaction}$, $N=\text{no reaction}$, $M=\text{mild reaction}$

There are 470 people under the event with no reaction and the total number people who participated in the study are 1000.

The value of $P\left(N\right)$ is as follows:

$\begin{array}{c}P\left(S\right)=\frac{470}{1,000}\\ =0.470\end{array}$

Thus, the value of $P\left(N\right)$ is 0.470.

The value of $P\left(M\right)$ is as follows:

$\begin{array}{c}P\left(M\right)=\frac{390}{1,000}\\ =0.390\end{array}$

Thus, the value of $P\left(M\right)$ is 0.390.

The value of $P\left(S\right)$ is computed as follows:

$\begin{array}{c}P\left(S\right)=\frac{140}{1,000}\\ =0.140\end{array}$

Therefore, the value of $P\left(S\right)$ is 0.140.

b.

To determine

Find the values of $P\left(N/W\right)$ and $P\left(S/W\right)$.

Answer to Problem 12CRP

The value of $P\left(N/W\right)$ is 0.84.

The value of $P\left(S/W\right)$ is 0.04.

Explanation of Solution

The value of $P\left(N/W\right)$ is computed as follows:

$\begin{array}{c}P\left(N/W\right)=\frac{P\left(NandW\right)}{P\left(W\right)}\\ =\frac{420}{500}\left\{\because \text{from the given table}\right\}\\ =0.84\end{array}$

Thus, the value of $P\left(N/W\right)$ is 0.84.

The value of $P\left(S/W\right)$ is as follows:

$\begin{array}{c}P\left(S/W\right)=\frac{P\left(SandW\right)}{P\left(W\right)}\\ =\frac{20}{500}\left\{\because \text{from the given table}\right\}\\ =0.04\end{array}$

Therefore, the value of $P\left(S/W\right)$ is 0.04.

c.

To determine

Find the values of $P\left(N/A\right)$ and $P\left(S/A\right)$.

Answer to Problem 12CRP

The value of $P\left(N/A\right)$ is 0.100.

The value of $P\left(S/A\right)$ is 0.240.

Explanation of Solution

The value of $P\left(N/A\right)$ is as follows:

$\begin{array}{c}P\left(N/A\right)=\frac{P\left(NandA\right)}{P\left(A\right)}\\ =\frac{50}{500}\left\{\because \text{from the given table}\right\}\\ =0.100\end{array}$

Thus, the value of $P\left(N/A\right)$ is 0.100.

The value of $P\left(S/A\right)$ is computed as follows:

$\begin{array}{c}P\left(S/A\right)=\frac{P\left(SandA\right)}{P\left(A\right)}\\ =\frac{120}{500}\\ =0.240\end{array}$

Therefore, the value of $P\left(S/A\right)$ is 0.240.

d.

To determine

Find the values of $P\left(NandW\right)$ and $P\left(MandW\right)$.

Answer to Problem 12CRP

The value of $P\left(NandW\right)$ is 0.420.

The value of $P\left(MandW\right)$ is 0.060.

Explanation of Solution

The value of $P\left(NandW\right)$ is as follows:

$\begin{array}{c}P\left(NandW\right)=P\left(W\right)\times P\left(N/W\right)\\ =\frac{500}{1,000}\times \frac{420}{500}\\ =\frac{420}{1,000}\\ =0.420\end{array}$

Thus, the value of $P\left(NandW\right)$ is 0.420.

The value of $P\left(MandW\right)$ is as follows:

$\begin{array}{c}P\left(MandW\right)=P\left(W\right)\times P\left(M/W\right)\\ =\frac{500}{1,000}\times \frac{60}{500}\\ =\frac{60}{1,000}\\ =0.060\end{array}$

Thus, the value of $P\left(MandW\right)$ is 0.060.

e.

To determine

Compute the value of $P\left(\text{N }or\text{ M}\right)$.

Explain whether the events $N=\text{no reaction}$ and $A=\text{mild reaction}$ are mutually exclusive.

Answer to Problem 12CRP

The value of $P\left(\text{N }or\text{ M}\right)$ is 0.860.

Explanation of Solution

Yes, the events $N=\text{no reaction}$ and $A=\text{mild reaction}$ are mutually exclusive, because these events cannot occur at the same time.

The value of $P\left(\text{N }or\text{ M}\right)$ is as follows:

$\begin{array}{c}P\left(\text{N }or\text{ M}\right)=P\left(N\right)+P\left(M\right)\\ =0.470+0.390\left\{\because \text{from Part}\left(a\right)\right\}\\ =0.860\end{array}$

Thus, the value of $P\left(\text{N }or\text{ M}\right)$ is 0.860.

f.

To determine

Explain whether the events $N=\text{no reaction}$ and $W=\text{washing oil off within 5 minute}$ are independent or not.

Answer to Problem 12CRP

The events N and W are not independent.

Explanation of Solution

The events N and W are said to be independent if $P\left(NandW\right)=P\left(N\right)\times P\left(W\right)$.

From Part (a), the value of $P\left(N\right)$ is 0.470.

From Part (d), the value of $P\left(W\right)$ is 0.500, and the value of $P\left(NandW\right)$ is 0.420.

The R.H.S $P\left(N\right)\times P\left(W\right)$ of the independent condition is as follows:   

$\begin{array}{c}P\left(N\right)\times P\left(W\right)=0.470\times 0.500\\ =0.235\\ \ne P\left(NandW\right)\\ \ne L.H.S\end{array}$

Since the condition $P\left(NandW\right)=P\left(N\right)\times P\left(W\right)$ is not satisfied. Therefore, the events N and W arenot independent.