There are 135 people in a sports centre
77 people use the gym
62 people use the swimming pool
65 people use the track
27 people use the gym and the pool
23 people use the pool and the track
31 people use the gym and the track
4 people use all three facilities
Given that a randomly selected person uses at least two facilities what is the probability they use all the facilities?
54
What’s the answer
To find the probability that a randomly selected person uses all three facilities, we need to calculate the number of people who use all three facilities and divide it by the total number of people who use at least two facilities.
From the information given, we know the following:
- 135 people in total
- 77 people use the gym
- 62 people use the swimming pool
- 65 people use the track
- 27 people use the gym and the pool
- 23 people use the pool and the track
- 31 people use the gym and the track
- 4 people use all three facilities
To find the number of people who use at least two facilities, we can add up the number of people using each individual facility and subtract the number of people using exactly one facility:
Number of people using at least two facilities = (Number of people using gym) + (Number of people using swimming pool) + (Number of people using track) - (Number of people using exactly one facility)
Number of people using at least two facilities = 77 + 62 + 65 - (27 + 23 + 31)
Number of people using at least two facilities = 181 - 81
Number of people using at least two facilities = 100
Now, to find the probability that a randomly selected person uses all three facilities, we divide the number of people who use all three facilities (4) by the total number of people who use at least two facilities (100):
Probability of using all three facilities = (Number of people using all three facilities) / (Number of people using at least two facilities)
Probability of using all three facilities = 4 / 100
Probability of using all three facilities = 0.04
Therefore, the probability that a randomly selected person uses all three facilities, given that they use at least two facilities, is 0.04 or 4%.
To find the probability that a randomly selected person uses all three facilities, we need to determine the total number of people who use at least two facilities and the number of people who use all three.
Let's break down the given information into a Venn diagram to visualize the situation:
```
Total People (135)
_______________________
/ \
/ \
Gym (77) / Pool (62) \ Track (65)
/ \
/_______________________________\
| |
| Gym and Pool (27) |
| Pool and Track (23) |
| Gym and Track (31) |
| |
| All three (4) |
|_______________________________|
```
From the Venn diagram, we can see that the total number of people who use at least two facilities is the sum of the following:
Number of people who use Gym and Pool = 27
Number of people who use Pool and Track = 23
Number of people who use Gym and Track = 31
Number of people who use all three = 4
So, the total number of people who use at least two facilities is 27 + 23 + 31 + 4 = 85.
Finally, we can calculate the probability by dividing the number of people who use all three facilities (4) by the total number of people who use at least two facilities (85):
Probability = Number of people who use all three / Total number of people who use at least two facilities
= 4 / 85
Therefore, the probability that a randomly selected person uses all three facilities given that they use at least two facilities is approximately 0.0471 (4/85).
Note: The probability is approximate since we assumed that the sample is representative of the larger population.