# Pairwise independence does not mean independence of events

This is taken from Stat 110

Give an example of $$3$$ events $$A, B, C$$ which are pairwise independent but not independent.

Hint: find an example where whether $$C$$ occurs is completely determined if we know whether $$A$$ occurred and whether $$B$$ occurred, but completely undetermined if we know only one of these things.

Solution (given): Consider two fair, independent coin tosses, and let $$A$$ be the event that the first toss is Heads, $$B$$ be the event that the second toss is Heads, and $$C$$ be the event that the two tosses have the same result.

I don't understand the hint properly. I do understand that the events are not independent since $$P(A|BC) \neq P(A)$$ (and similarly for $$B,C$$). But how exactly do the following statements:

1. whether $$C$$ occurs is completely determined if we know whether $$A$$ occurred and whether $$B$$ occurred
2. whether $$C$$ occurs is completely undetermined if we know of only one of $$A$$ and $$B$$

correspond to (in)dependence of the events?

Events are independent if, for a particular event, the occurrence of one or more of the other (remaining) events does not change the probability of occurrence of that particular event.

• By definition $n$ events are mutually independent, when each event is independent of any combination of other events. Commented Aug 22, 2021 at 22:05
• "completely undetermined if we know of only one of $A$ and $B$" means that $C$ is independent of $A$, and $C$ is independent of $B$, hence the three events are pairwise independent.
– Joe
Commented Aug 22, 2021 at 22:27

$$P(C|A) = \frac{P(A\cap C)}{P(A)}=\frac{P(HH)}{P(H)}=1/2=P(C)$$
• No like, how does A,B determining C correspond to $P(C|AB) \neq P(C)$? I don't get it, it doesn't feel natural Commented Aug 23, 2021 at 1:44
• @dictatemetokcus awesome, in this case we can even say that $P(C|A,B)=1$ Commented Aug 23, 2021 at 2:02