Multiplication of Probabilities
🎯 Big Idea
When we want to find the probability that two or more events both happen, we multiply their probabilities. But how we multiply depends on whether the events affect each other or not.
Independent Events
Definition: Two events $A$ and $B$ are independent if the occurrence of one does not affect the probability of the other.
Multiplication Rule for Independent Events
$$P(A \text{ and } B) = P(A) \times P(B)$$
Examples of independent events:
- Flipping a coin AND rolling a die
- Drawing a card, replacing it, then drawing again
- Two different students' test scores
Key phrase: "with replacement" usually signals independence.
Dependent Events
Definition: Two events $A$ and $B$ are dependent if the occurrence of one changes the probability of the other.
General Multiplication Rule
$$P(A \text{ and } B) = P(A) \times P(B | A)$$
Here, $P(B|A)$ is read as "the probability of $B$ given that $A$ has occurred" — this is called conditional probability.
Examples of dependent events:
- Drawing cards without replacement
- Picking two students from a class for a team
- Selecting marbles from a bag without putting them back