Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I've found the following exercise:

Suppose $A$,$B$ and $C$ are independent events and $P(A\cap B) \ne 0$. Show $P(C |A\cap B) = P(C)$

I've tried several times with no success and I would really appreciate some help.

share|cite|improve this question
up vote 0 down vote accepted

By the definition of conditional probability, $$\Pr(C|A\cap B)=\frac{\Pr(A\cap B\cap C)}{\Pr(A\cap B)}.$$ By independence $\Pr(A\cap B\cap C)=\Pr(A)\Pr(B)\Pr(C)$ and $\Pr(A\cap B)=\Pr(A)\Pr(B)$.

share|cite|improve this answer

Another solution is almost the same as the one by Andre, but perhaps the following fact would be useful for you.

If $A,B,C$ are mutually independent then $C\perp A\cap B$. Indeed: $$ P(C\cap (A\cap B)) =P(C\cap A\cap B) = P(A)P(B)P(C) = P(C)P(A\cap B). $$

Now, if $F\perp G$ then $P(F|G) = P(F)$, so that choose $F=C$ and $G = A\cap B$ which gives you $$ P(C|A\cap B) = P(C),. $$

share|cite|improve this answer
I've not seen $\perp$ used before in this context, but I like it. Am assuming it denotes independence, is that right? Do you know of any probability references that use it? – Assad Ebrahim May 19 '14 at 8:09
@AssadEbrahim: Yes, this means independence. I do remember using it back in Russia, however I am not sure I can recall precisely whether I saw such notation in English books on probability. I've seen in Kallenberg's Foundations they use of a similar symbol for conditional independence though. – Ilya Jun 4 '14 at 7:20

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.