Question

In a class of 30 students, 23 play an instrument and 12 play a sport. There are 3 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

Let I = plays an instrument, S = plays a sport. We have |I| = 23, |S| = 12, and 3 play neither, so |I ∪ S| = 30 − 3 = 27.

Use |I| + |S| − |I ∩ S| = |I ∪ S|:
23 + 12 − |I ∩ S| = 27 ⇒ |I ∩ S| = 8.

Thus P(I | S) = |I ∩ S| / |S| = 8/12 = 2/3 ≈ 66.7%.