Integrated Math 2 / Unit 9: Probability
9.3 Independent and Dependent Events
Notes
Two events are independent if one happening does not change the probability of the other. Otherwise they are dependent.
- Independent: $P(A \text{ and } B) = P(A) \cdot P(B)$.
- Dependent: $P(A \text{ and } B) = P(A) \cdot P(B \mid A)$.
- Drawing with replacement gives independent events. Drawing without replacement gives dependent events.
- Example: a bag has 5 red and 3 blue marbles. Two reds without replacement: $\displaystyle \frac{5}{8} \cdot \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$.
Practice
A fair coin is tossed and a fair die is rolled. What is $P(\text{heads and a 4})$?
The events are independent: $\displaystyle \frac{1}{2} \cdot \frac{1}{6} = \frac{1}{12}$.
A bag has 5 red and 3 blue marbles. Two are drawn without replacement. What is the probability both are red?
$\displaystyle \frac{5}{8} \cdot \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$.
Which pair of events is independent?
Separate die rolls do not affect each other. The other situations change what is left to choose from.
Ready for more? Take the Unit 9 test.