Question 41·Medium·Probability and Conditional Probability
A survey of 120 high school students recorded whether each student plays on a sports team and whether each student attends after-school music lessons.
| Attends music lessons | Does not attend music lessons | |
|---|---|---|
| Plays sports | 30 | 50 |
| Does not play sports | 20 | 20 |
If one of the students who plays sports is selected at random, what is the probability that this student also attends music lessons?
For conditional probability questions, first identify the "given" condition and restrict your total (denominator) to only that group. Then, within that restricted group, count how many satisfy the additional condition (numerator), form the fraction, and simplify. Be careful not to use the overall total from the table when the question clearly specifies a subgroup like "students who play sports."
Hints
Focus on the condition
The phrase "If one of the students who plays sports is selected" means you should only look at the row for students who play sports, not the entire table.
Determine the new total
How many students in total play sports (both those who attend music lessons and those who do not)?
Form the probability
Out of the students who play sports, how many also attend music lessons? Put that number over the total number of students who play sports and then simplify the fraction.
Desmos Guide
Compute the unsimplified probability
In Desmos, type 30/80 and look at the result it gives. This represents the probability before simplifying.
Read the simplified fraction
Desmos will automatically simplify 30/80 to a reduced fraction. The simplified fraction shown is the probability that a randomly chosen student who plays sports also attends music lessons.
Step-by-step Explanation
Identify the relevant group (the conditional part)
The question says: "If one of the students who plays sports is selected..." This means we only look at the students who play sports, not all 120 students.
From the table, the number of students who play sports is:
- Attends music lessons and plays sports: 30
- Does not attend music lessons and plays sports: 50
So the total number of students who play sports is .
Find how many in this group attend music lessons
Now focus on the students who both play sports and attend music lessons.
From the table, that number is 30.
So, out of the 80 sports-playing students, 30 also attend music lessons.
Write and simplify the probability
The probability we want is:
- Favorable outcomes: student plays sports and attends music lessons .
- Total possible outcomes (given they play sports): student plays sports .
So the probability is
Now simplify by dividing numerator and denominator by 10:
So the correct answer is .