There are 130 people in a sports center. Of these, 61 use the gym, 68 use the swimming pool, and 60 use the track. Additionally, 25 people use both the gym and the pool, 30 use both the pool and the track, and 19 use both the gym and the track. Six people use all three facilities. If a person is selected at random, what is the probability that they use exactly one of the facilities?
Mathematics · Middle School · Thu Feb 04 2021
Answered on
To find the probability that a person uses exactly one of the facilities, we first need to find the total number of people using exactly one facility and then divide that by the total number of people in the sports center.
Let's denote the number of people using each facility as follows: - G = Gym users = 61 - P = Pool users = 68 - T = Track users = 60 - GP = Both gym and pool users = 25 - PT = Both pool and track users = 30 - GT = Both gym and track users = 19 - GPT = All three facilities = 6
Using the principle of inclusion-exclusion, we can find the number of people using exactly one facility. This principle addresses the problem of double or triple counting people who are in multiple groups.
The formula for the number of people using exactly one facility is: G_only + P_only + T_only = G + P + T - (GP + PT + GT) + (2 * GPT)
Where: - G_only is the number of people using only the gym, - P_only is the number of people using only the pool, and - T_only is the number of people using only the track.
Now we can calculate:
G_only = G - (GP + GT) + GPT = 61 - (25 + 19) + 6 = 61 - 44 + 6 = 23 P_only = P - (GP + PT) + GPT = 68 - (25 + 30) + 6 = 68 - 55 + 6 = 19 T_only = T - (GT + PT) + GPT = 60 - (19 + 30) + 6 = 60 - 49 + 6 = 17
Total number of people using exactly one facility: G_only + P_only + T_only = 23 + 19 + 17 = 59
Total number of people in the sports center is 130.
To find the probability (P) that a person uses exactly one of the facilities, we use the formula:
P(exactly one facility) = Number of people using exactly one facility / Total number of people
P(exactly one facility) = 59 / 130
Now compute the probability:
P(exactly one facility) ≈ 0.4538 or 45.38%