# An example of a case where events A, B, C are pairwise independent, but not independent

This question is from Harvard's free STAT110 course:

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.

The sample answer given is similar to what I had in mind, but I'm curious if my case works. Suppose we flip two fair coins.

• $P(A) = \frac{1}{2} =$ first coin is heads (H)
• $P(B) = \frac{1}{2} =$ second coin is heads (H)
• $P(C) = \frac{1}{2} =$ the two coins we obtain are either HH, or TT

Checking:

• $P(A \cap B) = P(A)P(B)$ because they're independent events.
• $P(A \cap C) = \frac{1}{4} = P(A)P(C)$ since, if we see the HH/HT/TH/TT outcomes as quadrants, there is only one case where $A$ is true (first one is heads) and $C$ is true (we obtain HH).
• $P(B \cap C) = \frac{1}{4}$ as above.
• $P(A \cap B \cap C) = \frac{1}{4} \neq P(A)P(B)P(C)$ since if we know $A$ and $B$, there is only one outcome for $C$ possible.

Is this right, or am I missing something here?

• No, it seems ok to me. The coin toss you choosed is the standard example to demonstrate this difference. – Jimmy R. Jan 5 '17 at 10:07