Exercise C.2.8
$\star$ Two events $A$ and $B$ are conditionally independent, given $C$, if
$$ \Pr\{A \cap B | C\} = \Pr\{A | C\} \cdot \Pr\{B | C\} $$
Give a simple but nontrivial example of two events that are not independent but are conditionally independent given a third event.
Two people use the same coin.
- A - person 1 tosses heads
- B - person 2 tosses heads
- C - the coin is biased (falls heads only) - happens one in three times
Thus:
$$ \Pr\{A\} = \frac 1 3 \cdot 1 + \frac 2 3 \cdot \frac 1 2 = \frac 2 3 \\ \Pr\{B\} = \frac 1 3 \cdot 1 + \frac 2 3 \cdot \frac 1 2 = \frac 2 3 \\ \Pr\{A \cap B\} = \frac 1 3 \cdot 1 + \frac 2 3 \cdot \frac 1 4 = \frac 1 2 $$
The two events are not independent, because:
$$ \frac 2 3 = \Pr\{A \cap B\} \neq \Pr\{A\} \cdot \Pr\{B\} = \frac 4 9 $$
However, they are conditionally independent, because:
$$ 1 = \Pr\{A \cap B | C \} = \Pr\{A | C\} \cdot \Pr\{B | C\} = 1 $$