# Three events A, B and C are defined in the sample space S. The events A and B are mutually exclusive and A and C are independent. What is P(A|C)?

## $P \left(A\right) = 0.2 , P \left(B\right) = 0.4$ and $P \left(A \cup C\right) = 0.7$

Jan 28, 2017

$P \left(A | C\right) = 0.2$

#### Explanation:

We use the definition of Conditional Probability;

$P \left(X | Y\right) = \frac{P \left(X \cap Y\right)}{P \left(Y\right)}$

Se we have

$P \left(A | C\right) = \frac{P \left(A \cap C\right)}{P \left(C\right)}$

We are also told that $A$ and $C$ are independent events

$\therefore P \left(A \cap C\right) = P \left(A\right) \cdot P \left(C\right)$

So:

$P \left(A | C\right) = \frac{P \left(A\right) \cdot P \left(C\right)}{P \left(C\right)}$
$\text{ } = P \left(A\right)$
$\text{ } = 0.2$