6.4 Conditional Probability

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Notes

The conditional probability $P(A \mid B)$ is the probability of $A$ given that $B$ has occurred.

$\displaystyle P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}$.
General multiplication rule: $P(A \text{ and } B) = P(B) \cdot P(A \mid B)$.
$A$ and $B$ are independent exactly when $P(A \mid B) = P(A)$, or equivalently $P(A \text{ and } B) = P(A)P(B)$.
Example: If $P(A \text{ and } B) = 0.12$ and $P(B) = 0.4$, then $\displaystyle P(A \mid B) = \frac{0.12}{0.4} = 0.3$.

$P(A \text{ and } B) = 0.12$ and $P(B) = 0.4$. What is $P(A \mid B)$?

Two cards are drawn without replacement from a standard deck. Given that the first is an ace, what is the probability the second is an ace?

$P(A) = 0.5$, $P(B) = 0.4$ and $P(A \text{ and } B) = 0.2$. Are $A$ and $B$ independent?