# What is the difference between mutually independent and pairwise independent events in probability theory?

Let us assume that a number is selected at random from $1, 2, 3$. We define

$$A = \{1, 2\},\quad B = \{2, 3\},\quad C = \{1, 3\}$$

Then are $A$, $B$ and $C$ mutually independent or pairwise independent or both?

I am confused between mutually vs pairwise independent.

Mutual independence: Every event is independent of any intersection of the other events.

Pairwise independence: Any two events are independent.

$A, B, C$ are mutually independent if $$P(A\cap B\cap C)=P(A)P(B)P(C)$$ $$P(A\cap B)=P(A)P(B)$$ $$P(A\cap C)=P(A)P(C)$$ $$P(B\cap C)=P(B)P(C)$$

On the other hand, $A, B, C$ are pairwise independent if $$P(A\cap B)=P(A)P(B)$$ $$P(A\cap C)=P(A)P(C)$$ $$P(B\cap C)=P(B)P(C)$$

I'm sure you can solve your problem now.

• If I understood correctly, I think these are pairwise independent but not mutually independent? – Sangram Sep 9 '16 at 15:03
• What is your $P(A\cap B)$ and what are your $P(A)$ and $P(B)$? – JDF Sep 9 '16 at 15:04
• Also, what exactly is the event $A$? – JDF Sep 9 '16 at 15:05
• Question says 'Define A={1, 2}'. I assume it means A is an event that either 1 or 2 is selected. – Sangram Sep 9 '16 at 15:07
• If that is so, calculate the probabilities and verify. By the way, $A\cap B$ means the event that $A$ happens and $B$ happens, just to be clear. – JDF Sep 9 '16 at 15:10