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?

Short answer

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

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

Step-by-step solution

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$2\%$.