Question

in a class of 24 students, 12 play an instrument and 11 play a sport. there are 9 students who play an instrument and also play a sport. what is the probability that a student who plays a sport also plays an instrument?

Explanation:

Step1: Identify relevant numbers

Let $A$ be the event of playing an instrument and $B$ be the event of playing a sport. We know $n(A) = 12$, $n(B)=11$, $n(A\cap B)=9$ and total students $n = 24$. We want to find $P(A|B)$.

Step2: Recall conditional - probability formula

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. Since $P(A\cap B)=\frac{n(A\cap B)}{n}$ and $P(B)=\frac{n(B)}{n}$, then $P(A|B)=\frac{n(A\cap B)}{n(B)}$.

Step3: Calculate the probability

Substitute $n(A\cap B) = 9$ and $n(B)=11$ into the formula. So $P(A|B)=\frac{9}{11}$.

Answer:

$\frac{9}{11}$