Question

in a class of 40 students, 25 students play a musical instrument, and 10 of those students are also part of the schools choir. if a randomly selected student is known to play a musical instrument, what is the conditional probability that they are also part of the choir?

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the probability that a student is in the choir, given that they play a musical instrument, is .

Explanation:

Identify the given information

We are given the following values from the problem statement:

• Total number of students in the class: \(40\)
• Number of students who play a musical instrument: \(25\)
• Number of students who play a musical instrument and are also in the choir: \(10\)

Define the events and target probability

Let \(M\) be the event that a randomly selected student plays a musical instrument.
Let \(C\) be the event that a randomly selected student is in the choir.

We want to find the conditional probability that a student is in the choir, given that they play a musical instrument. This is denoted as \(P(C \mid M)\).

Apply the conditional probability formula

The formula for conditional probability is:

$$P(C \mid M) = \frac{P(C \cap M)}{P(M)}$$

Alternatively, using the counts of students:

$$P(C \mid M) = \frac{n(C \cap M)}{n(M)}$$

Where:

• \(n(C \cap M) = 10\) (students who play an instrument and are in the choir)
• \(n(M) = 25\) (students who play an instrument)

Calculate the final probability

Substitute the values into the formula:

$$P(C \mid M) = \frac{10}{25}$$

Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, \(5\):

$$P(C \mid M) = \frac{2}{5} = 0.4$$

Answer:

The probability that a student is in the choir, given that they play a musical instrument, is <blank>\(\frac{2}{5}\)</blank>.