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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.

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  • $\begingroup$ By definition $n$ events are mutually independent, when each event is independent of any combination of other events. $\endgroup$
    – zkutch
    Commented Aug 22, 2021 at 22:05
  • $\begingroup$ "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. $\endgroup$
    – Joe
    Commented Aug 22, 2021 at 22:27

1 Answer 1

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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.

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  • $\begingroup$ So it means, the events are independent pairwise. What does "whether C occurs is completely determined if we know whether A occurred and whether B occurred" mean then? I thought I understood this part, but now it seems, I didn't. (Sorry for that, I have edited my question.) $\endgroup$ Commented Aug 23, 2021 at 1:08
  • $\begingroup$ If A and B occur then we have the outcome “HH” which means C has occurred too (by definition) $\endgroup$
    – Annika
    Commented Aug 23, 2021 at 1:30
  • $\begingroup$ 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 $\endgroup$ Commented Aug 23, 2021 at 1:44
  • $\begingroup$ Ah, I get it, nevermind. Thank you. $\endgroup$ Commented Aug 23, 2021 at 1:46
  • $\begingroup$ @dictatemetokcus awesome, in this case we can even say that $P(C|A,B)=1$ $\endgroup$
    – Annika
    Commented Aug 23, 2021 at 2:02

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