There are 100 people in a sport centre. 67 people use the gym. 62 people use the swimming pool. 56 people use the track. 38 people use the gym and the pool. 31 people use the pool and the track. 33 people use the gym and the track. 16 people use all three facilities. Given that a randomly selected person uses the gym and the track, what is the probability they do not use the swimming pool?

We need to determinate who doesn't use anything, so first we must know the real number of who are practising: a person can do more things.

For doing this, is better use a three-circles graphic, but we try to do without it.

So

16 pool + gym + track, and this number is sure38 gym + pool, but this number includes the people of 1st point. We must know who do only gym + pool: 38 - 16 = 2231 pool + track, this number also includes people of 1st point, so: only pool + track 31 - 16 = 1533 gym + track, this number also includes people of 1st point, so: only gym + track 33 - 16 = 17

now:

67 gym, we must know who use only gym: 67 - 16 (gym + pool + track) - 22 (gym + pool) - 17 (gym + track) = 1262 pool, we must know who use only pool: 62 - 16 (gym + pool + track) - 22 (gym + pool) - 15 (pool + track) = 956 track, we must know who use only track: 56 - 16 (gym + pool + track) - 15 (pool + track) - 17 (gym + track) = 8

now we must now who do some facility: let's sum all

16 + 22 + 15 + 17 + 12 + 9 + 8 = 99

99/100 practise something

So only 1/100 doesn't use nothing

The probability is 1%

The answer that i got was 8/3