Statistics / Unit 6: Probability
6.3 The Multiplication Rule and Independence
Notes
Two events are independent if knowing one occurred does not change the probability of the other.
- If $A$ and $B$ are independent, $P(A \text{ and } B) = P(A) \cdot P(B)$.
- For events in a row, multiply the probabilities (when independent).
- Example: A coin lands heads and a die lands 6: $\displaystyle \frac{1}{2} \cdot \frac{1}{6} = \frac{1}{12}$.
- Mutually exclusive events with positive probabilities are never independent, since one occurring means the other cannot.
Practice
A coin is flipped and a die is rolled. What is the probability of heads and a 6?
Independent events: $\displaystyle \frac{1}{2} \cdot \frac{1}{6} = \frac{1}{12}$.
A basketball player makes $80\%$ of free throws, each shot independent of the others. The probability she makes two in a row is:
$0.8 \cdot 0.8 = 0.64$.
Events $A$ and $B$ are mutually exclusive and both have positive probability. Then $A$ and $B$ are:
$P(A \text{ and } B) = 0$, which is not equal to $P(A) \cdot P(B) > 0$.
Ready for more? Take the Unit 6 test.