Question 39·Hard·Probability and Conditional Probability
A health survey of 900 adults recorded whether each participant smoked and whether they exercised at least three times per week. Some of the survey results are shown in the table.
| Exercised regularly | Did not exercise regularly | Total | |
|---|---|---|---|
| Smoker | 180 | ||
| Non-smoker | 360 | 240 | |
| Total | 900 |
Overall, 30% of the participants were smokers.
If a participant who did not exercise regularly is selected at random, what is the probability that the participant was a smoker?
For conditional probability questions with tables, first fill in any missing counts using given percentages or totals so the table is complete. Then carefully identify the "given" condition (for example, people who did not exercise regularly) and restrict your denominator to just that row or column. Finally, form the probability as (number in the desired subcategory) ÷ (total in the given group), reduce if possible, and then convert to a decimal to compare quickly with the answer choices.
Hints
Use the 30% information
You know that 30% of the 900 participants were smokers. How many people is that?
Solve for x in the table
In the smoker row, the 180 who exercised plus the who did not exercise must add up to the total number of smokers you just found. Write and solve that equation.
Focus on the non-exercisers column
After you find , add it to 240 to get the total number of people who did not exercise regularly. Then think: among these people, how many are smokers?
Set up the probability correctly
The probability should be (smokers who did not exercise) divided by (total who did not exercise). Make sure your denominator is only people who did not exercise regularly, not all 900 participants.
Desmos Guide
Find the number of smokers and non-exercising smokers
In Desmos, type 0.30*900 to find the total number of smokers (270). Then type 270-180 to find how many smokers did not exercise regularly (this is ).
Find the total number of non-exercisers
In Desmos, type x+240 but replace x with the value you just found (for example, 90+240) to get the total number of people who did not exercise regularly.
Compute the conditional probability
In Desmos, type (smokers_who_did_not_exercise)/(total_who_did_not_exercise) using your two previous results (for example, 90/330). The decimal output shown by Desmos is the probability you should match to one of the answer choices.
Step-by-step Explanation
Find the total number of smokers
We are told that 30% of the 900 participants were smokers.
Calculate the number of smokers:
So there were 270 smokers in total.
Find how many smokers did not exercise regularly
In the smoker row, 180 smokers exercised regularly and smokers did not exercise regularly.
The row total for smokers must be 270, so:
Subtract 180 from both sides:
So 90 smokers did not exercise regularly.
Find the total number of people who did not exercise regularly
The column for people who did not exercise regularly has:
- 90 smokers (we just found )
- 240 non-smokers
So the total number of people who did not exercise regularly is:
Compute the conditional probability
We are choosing from the group that did not exercise regularly, so our "universe" is the 330 non-exercisers.
The probability that a randomly chosen non-exerciser is a smoker is
Simplify the fraction:
So the correct answer is 0.27.