Probabilities qns....?

Question

In a certain sample space, the events A and B are independent and P(A U B) = 5/8 and P(A and B') = 7/24. Calculate

(a) P(B)

My solutions:

P(A|B) = P(A and B) / P(B)

P(A|B) = 17/25 / P(B)

I tried to use the formula... but was somehow stucked..

.__.

(b) P(A)

(c) P(A and B)

(d) P(A' U B')

Since the first part I am already stucked, the rest I simply don't know how to do too...

Answer 1

(a) Calculate P(B).

We know that:

P(A) = P(A ∩ B) + P(A ∩ B')

P(A ∩ B') = P(A) - P(A ∩ B)

We also know that:

P(A U B) = P(A) + P(B) - P(A ∩ B)

P(A U B) = P(A) - P(A ∩ B) + P(B)

Use substitution.

P(A U B) = P(A ∩ B') + P(B)

P(B) = P(A U B) - P(A ∩ B') = 5/8 - 7/24 = 8/24 = 1/3

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(b) Calculate P(A).

Since events A and B are independent:

P(A) * P(B) = P(A ∩ B)

P(A) * P(B) = P(A) - P(A ∩ B')

P(A) [P(B) - 1] = - P(A ∩ B')

P(A) = -P(A ∩ B') / [P(B) - 1]

P(A) = P(A ∩ B') / [1 - P(B)] = (7/24) / [1 - 1/3] = (7/24) / (2/3) = 7/16

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(c) Calculate P(A ∩ B).

Since events A and B are independent:

P(A ∩ B) = P(A) * P(B) = (7/16) * (1/3) = 7/48

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(d) Calculate P(A' U B').

P(A' U B') = 1 - P(A ∩ B) = 1 - 7/48 = 41/48