Question

Suppose A and B are two events such that P(A) = 0.5 and P(A U B) = 0.8. Find P(B) assuming A and B are independent. I know the answer is 0.6 but I don't understand how to find P(B).

Answer 1

OK, I think I see how we can find it, I'm finishing up the solution.

Answer 2

Alright, thank you :)

Answer 3

We will need two equations. The first is the definition of P(A U B) and the second is \[P(A \cap B)\] when A and B are independent.

Answer 4

When we look these up, we know that \[P(A \cup B) = P(A) + P(B) - P(A \cap B)\] and \[P(A \cap B) = P(A) \times P(B)\] when A and B are independent.

Answer 5

Is that OK so far?

Answer 6

Yeah

Answer 7

Alright, now, what do we know? We know that P(A) = 0.5, and P(A U B) = 0.8. So, when we substitute these values we get this for the first equation: \[0.8 = 0.5 + P(B) - P(A \cap B)\]

Answer 8

Okay, I'm understanding so far.

Answer 9

Great! And this for the second: \[P(A \cap B) = 0.5 \times P(B)\]

Answer 10

Aha! This is the moment of victory! Now, we put the two equations together and get \[0.8 = 0.5 + P(B) - 0.5 P(B)\]

Answer 11

And it follows easily that P(B) = 0.6 by solving... :)

Answer 12

alright, I think I got it. Thanks so much! :)

Answer 13

No problem! Good luck!