In a survey that asked 69 people to indicate the reasons why they do not go to the gym, 44 responded that they were too busy, 26 said they felt too tired/lazy to go, 12 said both.

Question

(a) If a participant is randomly selected from the study, determine the probability that he/she does not go to the gym because he/she too busy or because he/ she is too tired/lazy.
(b) If a participant is randomly selected from the study, determine the probability that he/she does not go to the gym for reasons not related to busyness or tiredness/laziness.

Answer 1

To solve this problem, we need to use the principle of inclusion-exclusion.

(a) The probability of not going to the gym because of busyness or tiredness/laziness can be calculated as follows:

P(busy or tired) = P(busy) + P(tired) - P(both)

P(busy) = 44/69
P(tired) = 26/69
P(both) = 12/69

Therefore, P(busy or tired) = 44/69 + 26/69 - 12/69 = 58/69

The probability that a participant does not go to the gym because they are too busy or too tired/lazy is 58/69.

(b) The probability of not going to the gym for reasons not related to busyness or tiredness/laziness is the complement of (a).

P(not busy or not tired) = 1 - P(busy or tired)

P(not busy or not tired) = 1 - 58/69 = 11/69

The probability that a participant does not go to the gym for reasons unrelated to busyness or tiredness/laziness is 11/69.