Chapter 13, Problem 13.45E

To determine

To find:The probability that a randomly selected high school student did not use any tobacco products.

Answer to Problem 13.45E

The probability that a randomly selected high school student did not use any tobacco products is 0.73.

Explanation of Solution

Given info:

The youth of U.S. has declined cigarette smoking in recent years while the use of some other tobacco products has increased. The high school students used several tobacco products in last 30 days, more than who had used more than half and multiple tobacco products. Let A, B and C denotes the events corresponding to the different types of tobacco products in last 30 days and these are given below:

Events A denotes cigarette, B denotes electronic cigarette and C denotes other tobacco products including cigars, pipes, smokeless tobacco and hookahs.

The probabilities that a randomly selected high school student used these different tobacco products are shown below:

$P\left(A\right)=0.09$, $P\left(B\right)=0.16$, $P\left(C\right)=0.12$, $P\left(A\text{and}B\right)=0.03$, $P\left(A\text{and}C\right)=0.04$, $P\left(B\text{and}C\right)=0.04$ and $P\left(A\text{and}B\text{and}C\right)=0.01$.

Calculation:

From the given information,

$P\left(A\right)=0.09$, $P\left(B\right)=0.16$, $P\left(C\right)=0.12$, $P\left(A\text{and}B\right)=0.03$, $P\left(A\text{and}C\right)=0.04$, $P\left(B\text{and}C\right)=0.04$ and $P\left(A\text{and}B\text{and}C\right)=0.01$.

The remaining probabilities are given below:

The probability of $\left(A\text{and}B\text{and}\text{not}C\right)$ is given as,

$\begin{array}{c}P\left(A\text{and}B\text{and}\text{not}C\right)=P\left(A\right)-\left[\begin{array}{l}P\left(A\text{and}B\right)\\ +P\left(A\text{and}C\right)\end{array}\right]\\ =0.09-\left[0.03+0.04\right]\\ =0.09-0.07\\ =0.02\end{array}$

The probability of $\left(A\text{and}\text{not}B\text{and}\text{not}C\right)$ is given as,

$\begin{array}{c}P\left(\begin{array}{l}A\text{and}\text{not}\\ B\text{and}\text{not}C\end{array}\right)=P\left(A\right)-\left[\begin{array}{l}P\left(A\text{and}C\right)+P\left(A\text{and}B\text{and}C\right)\\ +P\left(A\text{and}B\text{and}\text{not}C\right)\end{array}\right]\\ =0.09-\left[0.03+0.01+0.02\right]\\ =0.09-0.06\\ =0.03\end{array}$

The probability of $\left(A\text{and}C\text{and}\text{not}B\right)$ is given as,

$\begin{array}{c}P\left(\begin{array}{l}A\text{and}C\\ \text{and}\text{not}B\end{array}\right)=P\left(A\text{and}C\right)-P\left(\begin{array}{l}A\text{and}B\\ \text{and}C\end{array}\right)\\ =0.04-0.01\\ =0.03\end{array}$

The probability of $\left(B\text{and}A\text{and}\text{not}C\right)$ is given as,

$\begin{array}{c}P\left(\begin{array}{l}B\text{and}C\\ \text{and}\text{not}A\end{array}\right)=P\left(B\text{and}C\right)-P\left(\begin{array}{l}A\text{and}\\ B\text{and}C\end{array}\right)\\ =0.04-0.01\\ =0.03\end{array}$

The probability of $\left(B\text{and}\text{not}A\text{and}\text{not}C\right)$ is given as,

$\begin{array}{c}P\left(B\text{and}\text{not}A\text{and}\text{not}C\right)=P\left(B\right)-\left[\begin{array}{l}P\left(A\text{and}B\text{and}C\right)\\ +P\left(A\text{and}B\text{and}\text{not}C\right)\\ +P\left(B\text{and}C\text{and}\text{not}A\right)\end{array}\right]\\ =0.16-\left[0.01+0.02+0.03\right]\\ =0.16-0.06\\ =0.10\end{array}$

The probability of $\left(C\text{and}\text{not}A\text{and}\text{not}B\right)$ is given as,

$\begin{array}{c}P\left(\begin{array}{l}C\text{and}\text{not}A\\ \text{and}\text{not}B\end{array}\right)=P\left(C\right)-\left[\begin{array}{l}P\left(A\text{and}B\text{and}C\right)\\ +P\left(A\text{and}C\text{and}\text{not}B\right)\\ +P\left(B\text{and}C\text{and}\text{not}A\right)\end{array}\right]\\ =0.12-\left[0.01+0.03+0.03\right]\\ =0.12-0.07\\ =0.05\end{array}$

Venn diagram:

The Venn diagram of the events A, B and C and mark the probabilities of all combinations of school students used different tobacco product is given below:

Probability:

The probability that a randomly selected high school student did not use any tobacco productsis obtained as shown below:

$\begin{array}{c}P\left(\text{No}\text{products}\right)=1-\left[\begin{array}{l}P\left(A\right)+P\left(B\right)+P\left(C\right)-\\ P\left(A\text{and}B\right)-P\left(B\text{and}C\right)\\ -P\left(A\text{and}C\right)+P\left(A\text{and}B\text{and}C\right)\end{array}\right]\\ =1-\left[\begin{array}{l}0.09+0.16+0.12-0.03-\\ 0.04-0.04+0.01\end{array}\right]\\ =1-0.27\\ =0.73\end{array}$

Thus, the probability that a randomly selected high school student did not use any tobacco products is 0.73.

Interpretation:

Among the different types of tobacco products used by the high school students, there are 73% students who do not use any tobacco products.