Chapter 5.3, Problem 21SE

a.

To determine

Find the probability $P\left(S^{\prime }\right)$.

Answer to Problem 21SE

The probability $P\left(S^{\prime }\right)$ is 0.754.

Explanation of Solution

Calculation:

The given information is that the event S represent the randomly chosen female aged 18-24 is a smoker and the event C represents that the randomly chosen female aged 18-24 is Caucasian. Also, $P\left(S\right)=0.246$, $P\left(C\right)=0.830$ and $P\left(S\cap C\right)=0.232$.

The formula for finding probability $P\left(S^{\prime }\right)$ is,

$P\left(S^{\prime }\right)=1-P\left(S\right)$

Substitute $P\left(S\right)=0.246$

$\begin{array}{c}P\left(S^{\prime }\right)=1-0.246\\ =0.754\end{array}$

Thus, the probability $P\left(S^{\prime }\right)$ is 0.754.

Hence, it can be concluded that there is a 75.4% $\left(=0.754\times 100\%\right)$ chance that the female aged 18-24 is a non-smoker.

b.

To determine

Find the probability $P\left(S\cup C\right)$.

Answer to Problem 21SE

The probability $P\left(S\cup C\right)$ is 0.844.

Explanation of Solution

Calculation:

The formula for finding probability $P\left(S\cup C\right)$ is,

$P\left(S\cup C\right)=P\left(S\right)+P\left(C\right)-P\left(S\cap C\right)$

Substitute $P\left(S\right)=0.246$, $P\left(C\right)=0.830$ and $P\left(S\cap C\right)=0.232$

$\begin{array}{c}P\left(S\cup C\right)=0.246+0.830-0.232\\ =0.844\end{array}$

Thus, the probability $P\left(S\cup C\right)$ is 0.844.

Hence, it can be concluded that there is an 84.4% $\left(=0.844\times 100\%\right)$ chance that the female aged 18-24 is a smoker or Caucasian.

c.

To determine

Find the probability $P\left(S|C\right)$.

Answer to Problem 21SE

The probability $P\left(S|C\right)$ is 0.2795.

Explanation of Solution

Calculation:

The formula for finding probability $P\left(S|C\right)$ is,

$P\left(S|C\right)=\frac{P\left(S\cap C\right)}{P\left(C\right)}$

Substitute $P\left(C\right)=0.830$ and $P\left(S\cap C\right)=0.232$

$\begin{array}{c}P\left(S|C\right)=\frac{0.232}{0.830}\\ =0.2795\end{array}$

Thus, the probability $P\left(S|C\right)$ is 0.2795.

Hence, it can be concluded that there is an 27.95% $\left(=0.2795\times 100\%\right)$ chance that the female aged 18-24 is smoker given that female aged 18-24 is a Caucasian.

d.

To determine

Find the probability $P\left(S|C^{\prime }\right)$.

Answer to Problem 21SE

The probability $P\left(S|C^{\prime }\right)$ is 0.0824.

Explanation of Solution

Calculation:

The formula for finding probability $P\left(S|C^{\prime }\right)$ is,

$\begin{array}{c}P\left(S|C^{\prime }\right)=\frac{P\left(S\cap C^{\prime }\right)}{P\left(C^{\prime }\right)}\\ =\frac{P\left(S\right)-P\left(S\cap C\right)}{1-P\left(C\right)}\langle \because P\left(S\cap C\right)+P\left(S\cap C^{\prime }\right)=P\left(S\right)\rangle \end{array}$

Substitute $P\left(S\right)=0.246$, $P\left(C\right)=0.830$ and $P\left(S\cap C\right)=0.232$

$\begin{array}{c}P\left(S|C^{\prime }\right)=\frac{0.246-0.232}{1-0.830}\\ =\frac{0.014}{0.17}\\ =0.0824\end{array}$

Thus, the probability $P\left(S|C^{\prime }\right)$ is 0.0824.

Hence, it can be concluded that there is an 8.24% $\left(=0.0824\times 100\%\right)$ chance that the female aged 18-24 is smoker given that female aged 18-24 is a not Caucasian.