Question

Data from Office on Smoking and Health, Centers for Disease Control and Prevention, indicate that 41% of adults who did not finish high school, 35% of high school graduates, 22% of adults who completed some college, and 15% of college graduates smoke. Suppose that one individual is selected at random and it is discovered that the individual smokes. Use the probabilities in the following table to calculate the probability that the individual is a college graduate.

Education Employed Unemployed
Not a high school graduate 0.0975 0.0080
High school graduate 0.3108 0.0128
Some college, no degree 0.1785 0.0062
Associate Degree 0.0849 0.0023
Bachelor Degree 0.1959 0.0041
Advanced Degree 0.0975 0.0015
Probability =

Hints: This problem has all the information you need, but not in the typical ready-to-use form. The table above can tell you the proportion of people with various levels of education in the population. Keep in mind that any degree (Associate, Bachelor, or Advanced) counts as graduating from college.

Answer 1

Probability ≈ 0.2771

Explanation:

Using the data provided, you want the probability that a smoker is a college graduate.

Conditional probability

The probability a smoker is a college graduate is the fraction of the population of smokers that are college graduates. To find this, we need to know the fraction of the population that smokes, and the fraction of the population that has graduated from college and smokes.

Education categories

We are given data regarding the fraction of the population in each education category that smokes. To find the fraction of the total population that represents, we need to know the fraction of the population in each education category.

We are given employment information for each education category. This can be used to find the population in each education category. That is, the fraction of the population that has not finished high school, for example, will be the sum of the fractions of that category who are employed and who are unemployed.

The attached spreadsheet uses the given employment data to find the population fraction in each education category. Multiplying that by the fraction of smokers in that category gives the fraction of the population that smokes.

For example, adding employment numbers for those who haven't finished high school, we see that 10.55% of the population falls in that category. Of those, 41% smoke, so about 4.33% of the total population are smokers who haven't finished high school.

Smokers

Using similar calculations for the other education categories, we find a total of about 25.51% of the population smokes, and about 5.79% of the population are college graduates who smoke.

Then our conditional probability is ...

P(college grad | smoker) = P(college grad & smoker) / P(smoker)

P(college grad | smoker) = 0.057930/0.255079 ≈ 0.2271

The probability a smoker is a college graduate is about 0.2271.

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Additional comment

The attached spreadsheet shows the results of the calculations described above.

The problem statement tells us that all with Associate degrees or higher are considered to be graduates from college. The employment numbers are broken out by the kind of college degree. In the spreadsheet, we have added all of the numbers together for the college degrees.

We find it interesting that the fraction of smokers decreases with education level. However, the largest fraction of the population is college graduates, so college graduates make up the second-largest group of smokers, after high school graduates.

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