# They satisfy the equation but they aren't independent events

I'm a little bit confused about independence of events. I've been reading about it,and doing some problems in order to get ready for my exam, but I still can't think of an example. The problem is to find a sample space with three events A,B,C that satisfy $$P(A\cap B\cap C)=P(A)P(B)P(C)$$ but that aren't independent.

But if they satisfy the equation, aren't they supposed to be independent? Is there a book I can read about it?

• 'But if they satisfy the equation, aren't they supposed to satisfy the equation' ????? You're saying exactly the same thing – Hippalectryon Sep 12 '14 at 5:33
• Yes, sorry, my bad. – Deni Sep 12 '14 at 5:46

## 2 Answers

Three events $A$, $B$, $C$ are independent if$$P(A\cap B\cap C)=P(A)P(B)P(C)$$ and if $$P(A\cap B)=P(A)P(B),\quad P(B\cap C)=P(B)P(C),\quad P(A\cap C)=P(A)P(C).$$ Thus you are asked to find $A$, $B$, $C$ such that the first condition holds but at least one of the others fail.

Note that if $P(C)=0$ and $P(A\cap B)\ne P(A)P(B)$, you are done. To be more ambitious, you could try to find a counterexample where $P(A)P(B)P(C)\ne0$.

I don't think the problem as stated is well defined. The definition of independence is that the probability of the joint event is the product of the probabilities of the individual events.