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

The total number of students is 28.

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

The number of students playing is 28 – 7 = 21

The number of students playing both games is 16 + 11 – 21 = 6

The number of students playing sport only is 11 – 6 = 5

The number of students playing instrument only is 16 - 6 = 10

Now, the number of students playing sport or instrument = 10 + 5 = 15

Let $A$ and $B$ be the events that a student plays instrument and a sport respectively then,

$p(A)={16\over28}\\p(B)={11\over28}$

The probability we need to determine is,

$p(A|B)={p(A\cap B)\over p(B)}$

We have that,

$p(A\cap B)={6\over28}\space and \space p(B)={11\over28}$

Thus,

$p(A|B)={{6\over28}\over{11\over28}}={6\over11}$

The probability that a student plays an instrument given that they play a sport is ${6\over11}$