Question

In a class of 28 students, 16 play an instrument and 11 play a sport. There are 7 students who do not play an instrument or a sport. What is the probability that a student plays an instrument given that they play a sport?

Answer 1

Number of students = 28

16 of them play an instrument and 11 of them play a sport. There are 7 students who do not play either instrument or a sport.

Number of students playing = 28 – 7 = 21

Number of students playing both games = 16 + 11 – 21 = 6

Number of students playing only instrument = 16 - 6 = 10

Number of students playing only sport = 11 – 6 = 5

Number of students playing instrument or sport = 10 + 5 = 15

Let $I$ and $S$ be the events that a student plays instrument and a sport respectively then,

$p(I)={16\over28}\\p(S)={11\over28}$

We find the probability,

$p(I|S)={p(I\cap S)\over p(S)}$

Now,

$p(I\cap S)={6\over28}\space and \space p(S)={11\over28}$

Therefore,

$p(I|S)={{6\over28}\over{11\over28}}={6\over11}$

Therefore, the probability that a student plays an instrument given that they play a sport is ${6\over11}$.