Algebra 2 / Unit 9: Probability
9.6 Conditional Probability
Notes
Conditional probability $P(B \mid A)$ is the probability of $B$ given that $A$ happened.
$$P(B \mid A) = \frac{P(A \text{ and } B)}{P(A)}$$
- With a two-way table, restrict to the row or column for the given condition, then divide.
- $P(B \mid A)$ and $P(A \mid B)$ are usually different.
- Example: of 40 students who own a dog, 10 also own a cat, so $\displaystyle P(\text{cat} \mid \text{dog}) = \frac{10}{40} = 0.25$.
Practice
60% of students play a sport, and 24% play a sport and an instrument. What is $P(\text{instrument} \mid \text{sport})$?
$\displaystyle \frac{0.24}{0.60} = 0.4$.
A school has 120 juniors and 80 seniors. 45 juniors and 50 seniors drive to school. What is $P(\text{drives} \mid \text{senior})$?
Only look at the 80 seniors: $\displaystyle \frac{50}{80} = 0.625$. The value 0.526 is $P(\text{senior} \mid \text{drives})$.
Two dice are rolled and the sum is 8. What is the probability that one of the dice shows a 3?
Sums of 8: (2,6), (3,5), (4,4), (5,3), (6,2). Two of these five have a 3.
Ready for more? Take the Unit 9 test.