there are 135 people in a sports center
73 people use the gym
73 people use the swimming pool
67 people use the track
36 people use the gym and the pool
35 people use the pool and the track
32 people use the gym and the track
14 people use all 3 facilities
a person is selected at random
what is the probablilty that this person doesnt use any facility
To find the probability that a person doesn't use any facility, we need to subtract the number of people using at least one facility from the total number of people.
First, let's find the number of people using at least one facility. We can use the principle of inclusion-exclusion to calculate this.
The number of people using at least one facility = (number of people using the gym) + (number of people using the swimming pool) + (number of people using the track)
- (number of people using both the gym and the pool) - (number of people using both the pool and the track) - (number of people using both the gym and the track)
+ (number of people using all three facilities)
Number of people using at least one facility = 73 + 73 + 67 - 36 - 35 - 32 + 14
Number of people using at least one facility = 224
The total number of people is given as 135.
So, the number of people not using any facility = Total number of people - Number of people using at least one facility
= 135 - 224
= -89 (which means it is not possible to have negative people)
Since there cannot be a negative number of people, the probability that a person doesn't use any facility is 0.
To find the probability that a person selected at random doesn't use any facility, we can use the concept of set theory and Venn diagrams.
First, let's represent the given information in a Venn diagram. Draw three overlapping circles to represent the gym, swimming pool, and track. Label the number of people in each intersection or area as given in the problem.
Based on the information provided:
- The number of people using the gym only is 73 - 36 - 32 + 14 = 19.
- The number of people using the swimming pool only is 73 - 36 - 35 + 14 = 16.
- The number of people using the track only is 67 - 35 - 32 + 14 = 14.
- The number of people using the gym and the swimming pool only is 36 - 14 = 22.
- The number of people using the gym and the track only is 32 - 14 = 18.
- The number of people using the swimming pool and the track only is 35 - 14 = 21.
To find the number of people not using any facility, we need to subtract the sum of people using at least one facility from the total number of people in the sports center:
Number of people not using any facility = Total number of people – (Sum of people using at least one facility)
Total number of people = 135
Sum of people using at least one facility = (Number of people using the gym only) + (Number of people using the swimming pool only) + (Number of people using the track only) + (Number of people using the gym and the swimming pool only) + (Number of people using the gym and the track only) + (Number of people using the swimming pool and the track only) + (Number of people using all three facilities)
Substituting the values we found earlier:
Sum of people using at least one facility = 19 + 16 + 14 + 22 + 18 + 21 + 14 = 124
Number of people not using any facility = 135 - 124 = 11
Therefore, the probability that a person selected at random doesn't use any facility is 11/135, which can be simplified or expressed as a decimal or percentage as needed.
Perfectly suited for a Venn diagram setup
Start with the intersection of 14, fill in that data in the intersection of all 3 circles that you must have drawn.
Now look at the intersection of any two circles:
I will pick the gym and track circles, it says that 36 use both of those
but the 14 from the intersection is included in that count, so 22 would go
in the part describing ONLY gym and track, so put 22 in that region.
Continue with the other two intersection of two circles
Now look at each of the whole circles.
e.g. the gym circle, it says that 73 use the gym.
Obviously that would include all those data values that you already
entered in the "gym" circle. So subtract those values form 73 and enter
the rest in the "gym only" part of the circle.
Continue for the other two circles.
Now add up all those values in the circles. If that total is less than 135
then the difference would go outside the 3 circles.
Your prob is that number / 135