Question

There are 135 people in a sport centre. 73 people use the gym. 59 people use the swimming pool. 31 people use the track. 19 people use the gym and the pool. 9 people use the pool and the track. 16 people use the gym and the track. 4 people use all three facilities. A person is selected at random. What is the probability that this person doesn't use any facility?

Answer 1

Answer:

Probability that this person doesn't use any facility is 0.09.

Step-by-step explanation:

We are given that there are 135 people in a sport center. 73 people use the gym. 59 people use the swimming pool. 31 people use the track. 19 people use the gym and the pool. 9 people use the pool and the track. 16 people use the gym and the track. 4 people use all three facilities.

Let Probability that people use the gym = P(G) = $\frac{73}{135}$ = 0.54

Probability that people use the swimming pool = P(S) = $\frac{59}{135}$ = 0.44

Probability that people use the track = P(T) = $\frac{31}{135}$ = 0.23

Probability that people use the gym and the pool = P($G \bigcap S$) = $\frac{19}{135}$ = 0.14

Probability that people use pool and the track = P($S \bigcap T$) = $\frac{9}{135}$ = 0.07

Probability that people use the gym and the track = P($G \bigcap T$) = $\frac{16}{135}$ = 0.12

Probability that people use all three facilities = P($G \bigcap S \bigcap T$) = $\frac{4}{135}$ = 0.03

Now, first we will find the probability of person using at least one of the facility, i.e.; $P(G \bigcup S \bigcup T)$

As, $P(G \bigcup S \bigcup T)$ = $P(G) + P(S) + P(T) - P(G \bigcap S) - P(G \bigcap S) - P(G \bigcap S) + P(G \bigcap S \bigcap T)$

So, $P(G \bigcup S \bigcup T)$ = $0.54 + 0.44+0.23-0.14-0.07-0.12+0.03$

                            = 1.24 - 0.33 = 0.91

Therefore, Probability that this person is using at least one of the facility is 0.91.

So, Probability that this person doesn't use any facility = 1 - Probability that this person is using at least one of the facility is 0.91

= 1 - 0.91 = 0.09

Hence, probability that this person doesn't use any facility is 0.09.

Answer 2

Answer:

Probability that the person doesn't use any facility = 0.0889

Step-by-step explanation:

Total number of people in a sports center = 135

Number of people using gym 'G' = 73

Number of people using swimming pool 'P' = 59

Number of people using track 'T' = 31

Number of people using gym and pool (G∩P) = 19

Number of people using pool and track (P∩T) = 9

Number of people using gym and track (G∩T) = 16

Number of people using all three facilities (G∩P∩T) = 4

Therefore, from the rule of compound probability,

P(GUPUT) = P(G) + P(P) + P(T) - P(G∩P) - P(G∩T) - P(P∩T) + p(G∩P∩T)

We know probability of an event = $\frac{\text{Favorable event}}{\text{Total outcomes}}$

We will plug in the values of probabilities of each event in the formula.

P(GUPUT) = $\frac{73}{135}+\frac{59}{135}+\frac{31}{135}-\frac{19}{135}-\frac{9}{135}-\frac{16}{135}+\frac{4}{135}$

                 = $\frac{1}{135}(73+59+31-19-9-16+4)$

                 = $\frac{123}{135}$

And probability that a person doesn't use any facility = 1 - P(GUPUT)

= 1 - $\frac{123}{135}$

= $\frac{135-123}{135}$

= $\frac{12}{135}$

= $\frac{4}{45}$ ≈ 0.0889