Q. 2.11 - A First Course in Probability

Question

A total of $28$ percent of American males smoke cigarettes, $7$ percent smoke cigars, and $5$ percent smoke both cigars and cigarettes.

(a)What percentage of males smokes neither cigars nor cigarettes?

(b)What percentage smokes cigars but not cigarettes?

Step-by-Step Solution

Answer

(a)The percentage of males smoke neither cigars nor cigarettes is $70\%$.

(b)The percentage of males smoke cigars but not cigarettes is $2\%$

Part (a) Step 1. Given information.

From the given information we have,

Probability of American males smoking cigarettes $=P\left(A\right) = 0.28$

Probability of American males smoking cigars $=P\left(B\right) = 0.07$

Probability of American males smoking both cigars and cigarettes $=P(A\cap B) = 0.05$

Part (a) Step 2. Find the percentage of males that smoke neither cigars nor cigarettes.

The percentage of males that smoke neither cigars nor cigarettes $=P\left(A^c\cup B^c\right)$

$P\left(A^c\cup B^c\right)=P{\left(A\cup B\right)}^c$

$$\begin{gathered} \mathrm{P}\left(\mathrm{A}\cup \mathrm{B}\right)=\mathrm{P}\left(\mathrm{A}\right)+\mathrm{P}\left(\mathrm{B}\right)-\mathrm{P}\left(\mathrm{A}\cap \mathrm{B}\right) \\ =0.28+0.07-0.05 \\ =0.30 \end{gathered}$$

$$\begin{gathered} P{\left(A\cup B\right)}^c=1-P\left(A\cup B\right) \\ =1-0.30 \\ =0.70 \end{gathered}$$

Therefore, the percentage of males that smoke neither cigars nor cigarettes is $70\%.$$$\begin{gathered} P{\left(A\cup B\right)}^c=1-P\left(A\cup B\right) \\ =1-0.30 \\ =0.70 \end{gathered}$$

Part (b) Step 1. Find the percentage of males that smoke cigars but not cigarettes.

The percentage of males that smoke cigars but not cigarettes $=\mathrm{P}\left(\mathrm{B}\cap {\mathrm{A}}^{\mathrm{c}}\right)$

$$\begin{gathered} \mathrm{P}\left(\mathrm{B}\cap {\mathrm{A}}^{\mathrm{c}}\right)=\mathrm{P}\left(\mathrm{B}\right)-\mathrm{P}(\mathrm{A}\cap \mathrm{B}) \\ =0.07-0.05 \\ =0.02 \end{gathered}$$

Therefore, the percentage of males that smoke cigars but not cigarettes is $2\%$.