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In a class of 23 students, 11 play an instrument and 5 play a sport. There are 3 students who play an instrument and also play a sport. What is the probability that a student does not play an instrument given that they play a sport?
Answer:

In a class of $23$ students, $11$ play an instrument and $5$ play a sport. There are $3$ students who play an instrument and also play a sport. What is the probability that a student does not play an instrument given that they play a sport? $\newline$ Answer:

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Probability of independent and dependent events

Full solution

Q. In a class of

23

students,

11

play an instrument and

5

play a sport. There are

3

students who play an instrument and also play a sport. What is the probability that a student does not play an instrument given that they play a sport?

\newline

Answer:

Identify Total Students: First, we need to identify the total number of students who play a sport. According to the problem, there are $5$ students who play a sport.
Determine Students Playing Both: Next, we need to determine how many of those students who play a sport also play an instrument. The problem states that there are $3$ students who play both an instrument and a sport.
Calculate Students Playing Only: To find the probability that a student does not play an instrument given that they play a sport, we subtract the number of students who play both an instrument and a sport from the total number of students who play a sport. This will give us the number of students who play a sport but do not play an instrument. $\newline$ Number of students who play only a sport $=$ Total number of students who play a sport $-$ Number of students who play both an instrument and a sport $\newline$ Number of students who play only a sport $= 5 - 3 = 2$
Calculate Probability: Now, we calculate the probability. The probability that a student does not play an instrument given that they play a sport is the number of students who play only a sport divided by the total number of students who play a sport. $\newline$ Probability = $\frac{\text{Number of students who play only a sport}}{\text{Total number of students who play a sport}}$ $\newline$ Probability = $\frac{2}{5}$

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