Question

As the dangers of smoking have become more widely known, clear class differences in smoking have emerged. British government statistics classify adult men by occupation as “managerial and professional” ($43\%$of the population), “intermediate” ($34\%$), or “routine and manual” ($23\%$). A survey finds that $20\%$of men in managerial and professional occupations smoke, $29\%$of the intermediate group smoke, and $38\%$in routine and manual occupations smoke.

(a) Use a tree diagram to find the percent of all adult British men who smoke.

(b) Find the percent of male smokers who have routine and manual occupations.

Short answer

(a)$P($Smoke$)=27.20\%$

(b)$P(\text{Routine}\mid \text{smoke})=32.13\mathrm{\%}$

Step-by-step solution

Part (a) Step 1: Given information

Given in the question that, as the dangers of smoking have become more widely known, clear class differences in smoking have emerged. British government statistics classify adult men by occupation as “managerial and professional” ($43\%$of the population), “intermediate” ($34\%$), or “routine and manual” ($23\%$). A survey finds that $20\%$of men in managerial and professional occupations smoke, $29\%$of the intermediate group smoke, and $38\%$in routine and manual occupations smoke.

We need to find the percent of all adult British men who smoke by using a tree diagram.

Part (a) Step 2: Explanation

Multiply the probabilities along the branches to find the chance of a smoker in each category:

$$\begin{gathered} P\left(\text{managerial and Smoke}\right)=43\mathrm{\%}\times 20\mathrm{\%} \\ =0.43\times 0.20 \\ =0.086 \\ =8.6\mathrm{\%} \end{gathered}$$

$$\begin{gathered} P\left(\text{Interm}P\right(\text{ediate and Smoke}\left)\right)=34\mathrm{\%}\times 29\mathrm{\%} \\ =0.34\times 0.29 \\ =0.0986 \\ =9.86\mathrm{\%} \end{gathered}$$

$$\begin{gathered} P\left(\text{Routine and Smoke}\right))=23\mathrm{\%}\times 38\mathrm{\%} \\ =0.23\times 0.38 \\ =0.0874 \\ =8.74\mathrm{\%} \end{gathered}$$

Add the corresponding probabilities:

$$\begin{gathered} P\left(\text{smoke}\right))=0.086+0.0986+0.0874 \\ =0.2720 \\ =27.20\mathrm{\%} \end{gathered}$$

Part (b)Step 1: Given information

As the risks of smoking have become more publicly recognized, there have been evident socioeconomic differences in smoking. Adult men in the United Kingdom are classified as "managerial and professional" (43 percent of the population), "intermediate" (34%), or "routine and manual" (23%). According to a poll, 20% of men in management and professional occupations smoke, 29% of men in intermediate occupations smoke, and 38% of men in routine and manual occupations smoke.

We need to figure out what percentage of men smokers work in routine and manual jobs.

Part (b) Step 2: Percent of male smokers 

Multiply the probabilities along the branches to find the chance of a smoker in each category:

$P($managerial and Smoke$)=43\%\times 20\%=0.43\times 0.20=0.086=8.6\%$

$P($intermediate and Smoke)$=34\%\times 29\%=0.34\times 0.29=0.0986=9.86\%$

$P($Routine and Smoke $)=23\%\times 38\%=0.23\times 0.38=0.0874=8.74\%$

Add the corresponding probabilities:

$P($smoke $)=0.086+0.0986+0.0874=0.2720=27.20\%$

Then we obtain

$$\begin{gathered} P(\text{Routine}\mid \text{smoke})=\frac{P\left(\text{Routine and Smoke}\right)}{P\left(\text{smoke}\right)} \\ =\frac{0.0874}{0.2720} \\ \approx 0.3213 \\ =32.13\mathrm{\%} \end{gathered}$$