Intuitive Understanding of Independence of 3 Events

For three events to be independent, knowledge of the others should not affect the probability of each event. For pairwise independence, knowing one other event tells you nothing. But knowing both other events might shift the odds. For example: draw 3 overlapping circles. Put a 1 in the 3 way overlap. Put 7 in each of the 2-way overlaps that are not in the center. Put a 1 in each of the regions belonging to exactly one set. Put a 7 in the complement of the union. The total is 32; each set contains 16, each 2-way overlap contains 8, so for example $A$ and $B$ are independent since $P(A) = \frac{1}{2}, P(B) = \frac{1}{2},$ and $P(A\cap B) = \frac{1}{4}$. Likewise for the other pairs, so we have full two way independence. But 3-way independence fails: $\frac{1}{8} = P(A)P(B)P(C) ≠ P(A\cap B \cap C) = \frac{1}{32}$.

Now what does this mean? Suppose I know $A$ is true. $P(C)$ is unaffected, since half of the elements of $A$ are in $C$.

Now uppose I know $A$ is true and $B$ is true. This makes $P(C)$ drop from $\frac{1}{2}$ to $\frac{1}{32}$. Knowing one of the other truth values told you nothing, but knowing both of them does give you information that changes the probability.