Integrated Math 2 / Unit 9: Probability
9.5 Conditional Probability
Notes
Conditional probability is the probability of $A$ given that $B$ has happened.
- $\displaystyle P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}$.
- The condition $B$ becomes the new sample space.
- Events are independent exactly when $P(A \mid B) = P(A)$.
- Example: $P(A \text{ and } B) = 0.12$ and $P(B) = 0.4$ give $P(A \mid B) = 0.3$.
Practice
$P(A \text{ and } B) = 0.12$ and $P(B) = 0.4$. What is $P(A \mid B)$?
$\displaystyle \frac{0.12}{0.4} = 0.3$.
Two aces are dealt from a deck without replacement. Given that the first card is an ace, what is the probability the second is an ace?
Three aces remain among 51 cards: $\displaystyle \frac{3}{51} = \frac{1}{17}$.
In a school, $60\%$ of students take Spanish and $24\%$ take both Spanish and band. What fraction of Spanish students also take band?
$\displaystyle P(\text{band} \mid \text{Spanish}) = \frac{0.24}{0.60} = 0.4$.
Ready for more? Take the Unit 9 test.