# Section 13.4: Calculated probabilities

## 13.4 Outline

1. Keno games
2. Independent events
1. independent
2. dependent
3. Probability of an intersection
1. multiplication property of probability
2. birthday problem
4. Probability of a union
2. probability of independent events
5. Drawing with and without replacement
6. Stochastic processes and tree diagrams

## 13.4 Essential Ideas

Multiplication Property of Probability: If events E and F are independent events, then we can find the probability of an intersection by

P(E and F ) = P(E ) times P(F )

Addition Property of Probability: For any events E and F, the probability of their union can be found by

P(E or F ) = P(E ) + P(F ) − P(E and F )

Drawing with replacement means choosing the first item, noting the result, and then replacing the item back into the sample space before selecting the second item.

Drawing without replacement means selecting the first item, noting the result, than then selecting a second item without replacing the first item.