Question

1. Given that PA ∪ B=5/9 and PA ∩ B'=8/15 , determine P(B) .

A country club has a membership of 500 members and operates facilities that include a swimming pool and a gymnasium. The club president would like to know how many members regularly use the facilities. A survey of the members indicates that 70% regularly use the swimming pool (S), 50% regularly use the gymnasium (G), and 5% use neither of these facilities regularly. Calculate the probability of the club members who use the facilities of the swimming pool and gymnasium P(S∩G).

Answer 1

Solution:

Question 1)

Given:

P( A ∪ B) = 5/9 and P(A ∩ B') = 8/15 ,

Find P(B)=.........?

Let's consider:

P(A ∩ B') = 8/15

P(A) - P(A ∩ B) = 8/15

Thus

P(A) - 8/15 = P(A ∩ B)

That is:

P(A ∩ B) = P(A) - 8/15

Now using addition rule of probability:

P( A ∪ B) = P(A) + P(B) - P( A ∩ B)

5/9 = P(A) + P(B) - [  P(A) - 8/15 ]

5/9 = P(A) + P(B) - P(A) + 8/15

5/9 = P(B) + 8/15

or

Question 2)

Given:

70% regularly use the swimming pool (S),

50% regularly use the gymnasium (G), and

5% use neither of these facilities regularly.

That is:

P(S) = 0.70

P(G) = 0.50

P( Neither S nor G) = P( S' ∩ G' ) = 0.05

We have to calculate the probability of the club members who use the facilities of the swimming pool and gymnasium

P(S ∩ G)=..........?

Let's consider:

P( S' ∩ G' ) = 0.05

Using following formula, we get:

P( S' ∩ G' ) = P(S U G)'

P( S' ∩ G' ) = 1 - P(S U G)

0.05 = 1 - [ P(S) + P(G) - P( S ∩ G) ]

0.05 = 1 - [ 0.70 + 0.50 - P( S ∩ G) ]

0.05 = 1 - [ 1.20 - P( S ∩ G) ]

0.05 = 1 - 1.20 + P( S ∩ G)

0.05 = -0.20 + P( S ∩ G)

0.05 + 0.20 = P( S ∩ G)

0.25 = P( S ∩ G)

P( S ∩ G) = 0.25