## Definition ==Conditional Probability== is the probability of an event $A$ occurring given that another event $B$ has already occurred. It is defined as: $ P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0 $ Conditioning effectively restricts the sample space to $B$ and measures how much of $B$ is also in $A$. > [!example]- Two dice are rolled. Given that the sum is at least 10, what is the probability the sum is 12? > Let $A = \{\text{sum} = 12\}$ and $B = \{\text{sum} \geq 10\}$. > The outcomes in $B$ are: $(4,6),(5,5),(6,4),(5,6),(6,5),(6,6)$, so $|B| = 6$ and $P(B) = 6/36$. > The only outcome in $A \cap B$ is $(6,6)$, so $P(A \cap B) = 1/36$. > $ P(A \mid B) = \frac{1/36}{6/36} = \frac{1}{6} \approx 0.167 $