Probability (don't understand this)?

Question

2 events A and B are such that P(A) = 1/4, P(A | B) = 1/2 and P(B | A) = 2/3.

(a) Find P(A and B) [A inverse 'U' B]

Solution:

P(B|A) * P(A) [<<< I don't understand this part -.-]

2/3 * 1/4 = 1/6

(b) Find P(B)

Solution:

P(B) = 1/3 [why is it 1/3..?]

Help me out, thanks. Explanation is highly needed and appreciated. =)

Answer 1

a) P(B|A) = P(A and B)/P(A) This is the formula for conditional probability.

Rearranging gives us P(A and B) = P(B|A) * P(A)

b) Similarly, P(A|B) = P(A and B)/P(B)

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

Answer 2

This is a rule of conditional probabilities:

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

not sure what its name is, or how to prove it, but might think of it as a definition:

P(B|A) is the probability that B happened given that we know that A happened. This is the probability that a and b happen, dividing out the chances that A happens. To say this another way, what are the chances that b happens if we know that a happens = what are the chance that both happen for the situation where A happens =

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

part b:

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

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