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:
- whether $C$ occurs is completely determined if we know whether $A$ occurred and whether $B$ occurred
- 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.
Completely undetermined just means C is independent of B or A by themselves:
$$P(C|A) = \frac{P(A\cap C)}{P(A)}=\frac{P(HH)}{P(H)}=1/2=P(C)$$
And similarly for B.