Question

How to solve this problem with a detailed explanation?

Let A and B be events with P(A) = a and P(B) = b.

If A and B are independent, what are P(A ∪ B), P(A ∩ B^c), and P(A|B)?

Now, assume A and B are mutually exclusive. What are P(A ∪ B), P(A ∩ B^c), and P(A|B)?

Answer 1

Solution:

Given,

=>A and B be events with P(A) = a and P(B) = b

(a)

Given,

=>A and B are independent events

Explanation:

=>If A and B are independent events then we can write P(A B) = P(A).P(B)

Finding P(A U B):

=>We know that, P(A U B) = P(A) + P(B) - P(A B)

=>P(A U B) = P(A) + P(B) - P(A).P(B)

=>P(A U B) = a + b - ab

Finding P(A B^c):

=>P(A B^c) = P(A).P(B^c)

=>P(A B^c) = P(A).(1 - P(B)) as P(B) + P(B^c) = 1

=>P(A B^c) = a(1-b)

Finding P(A | B):

=>We know that P(A | B) = P(A B)/P(B)

=>P(A | B) = P(A).P(B)/P(B)

=>P(A | B) = P(A)

=>P(A | B) = a

(b)

Given,

=>A and B events are mutually exclusive.

Explanation:

=>As events A and B are mutually exclusive so P(A B) = 0

Finding P(A U B):

=>We know that, P(A U B) = P(A) + P(B) - 0

=>P(A U B) = P(A) + P(B)

=>P(A U B) = a + b

Finding P(A B^c):

=>P(A B^c) = P(A) - P(A B)

=>P(A B^c) = P(A) - 0

=>P(A B^c) = a

Finding P(A | B):

=>We know that P(A | B) = P(A B)/P(B)

=>P(A | B) = 0/P(B)

=>P(A | B) = 0

I have explained each and every part with the help of statements attached to it.