Question

1. a suburban high school has a population of 1376 students. the number of students who participate in sports is 649. the number of students who participate in music is 433. if the probability that a student participated in either sports or music is $\frac{974}{1376}$, what is the probability that a student participates in both sports and music?

Explanation:

Step1: Recall the formula for probability of union

Let \(P(S)\) be the probability of participating in sports, \(P(M)\) be the probability of participating in music, and \(P(S\cap M)\) be the probability of participating in both. The formula for \(P(S\cup M)\) is \(P(S\cup M)=P(S)+P(M)-P(S\cap M)\).
We know that \(n(S) = 649\), \(n(M)=433\), \(n = 1376\) and \(P(S\cup M)=\frac{974}{1376}\). First, find \(P(S)=\frac{n(S)}{n}=\frac{649}{1376}\) and \(P(M)=\frac{n(M)}{n}=\frac{433}{1376}\).

Step2: Rearrange the formula to solve for \(P(S\cap M)\)

From \(P(S\cup M)=P(S)+P(M)-P(S\cap M)\), we can get \(P(S\cap M)=P(S)+P(M)-P(S\cup M)\).
Substitute the values: \(P(S)+P(M)=\frac{649}{1376}+\frac{433}{1376}=\frac{649 + 433}{1376}=\frac{1082}{1376}\).
Then \(P(S\cap M)=\frac{1082}{1376}-\frac{974}{1376}\).

Step3: Calculate the value of \(P(S\cap M)\)

\(P(S\cap M)=\frac{1082 - 974}{1376}=\frac{108}{1376}=\frac{27}{344}\)

Answer:

\(\frac{27}{344}\)