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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 鈥渕anagerial and professional鈥� ($43\%$of the population), 鈥渋ntermediate鈥� ($34\%$), or 鈥渞outine and manual鈥� ($23\%$). 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.

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

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

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

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

Add the corresponding probabilities:

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

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.

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

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

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

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

Add the corresponding probabilities:

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

Then we obtain

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