In probability, does P(A) = P(AB) + P(AB')?

$\begingroup$

Let A and B be events.
It seems to make sense to be that the probability that A occurs is equal to the probability that A and B both occur PLUS the probability that A occurs and B does not. I haven't been able to find this identity in my textbook though.

$\endgroup$ 1

2 Answers

$\begingroup$

$$P(A)=P(A\Omega)=P(A\cap (B\cup B'))=P((A\cap B)\cup (A\cap B'))=P(AB)+P(AB'). $$

Where the last equality stands because AB and AB' have no intersections.

$\endgroup$ $\begingroup$

Your intuition is correct. If you write it is conditional probability, this should be apparent. Note that $P(AB) = P(A|B)P(B)$ and refer to Bayes conditional probability rule.

$\endgroup$

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy