5 pts Conditional Probability Let be a countable set and P

(5 pts) Conditional Probability. Let be a countable set and P : 2 R be a probability distribution. Furthermore, we define the conditional probability of the event A C given the event B C as P(AP) :-BB). Your task is to formally prove that P(AB) is a probability distribution dhe avent B C asAYur taok is to formaly prove

Solution

S be the sample space

A, B are two events from the sample space S

n(AB) denotes the compound events of A and B

The probability of A B is P(A B)=^{n(A UB)}/_n(B)

The conditional probability of the event A will occur given that B has occurred is given by

P(A/B)=( ^{n(A UB)}/_n(B) )/(n(B)/n(s))

^{n(A UB)}/_{n(B) =P(AB)}

(n(B)/n(s) =n(B)

P(A/B)   = P(A B)/P(B)