Question 44·Medium·Probability and Conditional Probability
| In Study-Abroad Program | Not in Study-Abroad Program | |
|---|---|---|
| Knows French | 18 | 8 |
| Does not know French | 12 | 22 |
The table above shows the results of a survey of students at a high school about whether they know French and whether they are enrolled in the school’s study-abroad program. One student is selected at random from those surveyed. If the student chosen knows French, what is the probability that the student is enrolled in the study-abroad program?
For table-based conditional probability questions, first circle or highlight the row or column that matches the given condition (here, "knows French"). Then, within that restricted group, the denominator is the total in that row/column, and the numerator is the count that also has the desired outcome. Write the probability as this fraction and simplify. Staying disciplined about "given" (denominator) and "asked for" (numerator) prevents mixing up rows/columns and avoids most errors.
Hints
Focus on the given condition
You are told that the chosen student knows French. Which row of the table does that tell you to focus on?
Decide what counts as total and favorable
Within the "Knows French" row, which number represents students in the study-abroad program, and which numbers together represent all students who know French?
Write the probability as a fraction
Write the probability as (number who know French and are in the program) over (total number who know French). Then simplify the fraction if possible.
Desmos Guide
Compute the conditional probability in Desmos
In the Desmos calculator, type 18/(18+8) and press Enter. The value that appears is the probability; if it is a decimal, you can tap it to see the corresponding fraction, which is the correct choice from the options.
Step-by-step Explanation
Identify the relevant group (given condition)
The question says the chosen student knows French. That means we should look only at the row labeled "Knows French" and ignore the "Does not know French" row.
From that row:
- Knows French and in study-abroad program: 18
- Knows French and not in study-abroad program: 8
So there are students who know French in total.
Set up the conditional probability
We want the probability that a student is in the study-abroad program, given that the student already knows French.
Conditional probability here is:
- Favorable outcomes: students who both know French and are in the study-abroad program, which is 18.
- Total possible outcomes (given condition): all students who know French, which is .
So the probability is
Compute and simplify the fraction
First add the total number of students who know French:
So the probability is
Now simplify by dividing numerator and denominator by their greatest common factor, 2:
So the correct answer is , which corresponds to choice B.