The multiplication rule tells you how to find the probability that two events both happen. It is used whenever a problem asks for P(A and B), such as drawing two cards, passing two tests, or having two parts both work. The key idea is that you multiply probabilities along a pathway.
This rule matters because many real situations happen in stages, not all at once.
If the events are independent, the first event does not change the probability of the second, so P(A and B) = P(A)P(B). If the events are dependent, the first event does change the probability of the second, so you must use conditional probability: P(A and B) = P(A)P(B|A). A branching probability diagram helps show this clearly because each branch represents a possible outcome and its probability.
To find the probability of a complete path, multiply the probabilities on the branches in that path.
Key Facts
- General multiplication rule: P(A and B) = P(A)P(B|A).
- For independent events: P(A and B) = P(A)P(B).
- P(B|A) means the probability that B happens given that A already happened.
- If A and B are independent, then P(B|A) = P(B).
- In a probability tree, multiply along a path to find the probability of that full sequence.
- For events without replacement, probabilities often change, so the events are usually dependent.
Vocabulary
- Multiplication rule
- A probability rule used to find the chance that two or more events all occur.
- Independent events
- Events are independent when the outcome of one event does not change the probability of the other event.
- Dependent events
- Events are dependent when the outcome of one event changes the probability of another event.
- Conditional probability
- Conditional probability is the probability of an event happening given that another event has already happened.
- Probability tree
- A probability tree is a branching diagram that shows possible outcomes and the probabilities of each stage.
Common Mistakes to Avoid
- Using P(A)P(B) for dependent events is wrong because the probability of B may change after A happens. Use P(A)P(B|A) when the events affect each other.
- Forgetting to adjust the denominator after an event without replacement is wrong because the total number of possible outcomes has changed. After drawing one card from a deck, there are 51 cards left, not 52.
- Adding probabilities along a single path is wrong because a path represents events happening together. Multiply probabilities along a path to find P(A and B).
- Confusing P(A and B) with P(A or B) is wrong because and means both events occur, while or means at least one event occurs. The multiplication rule is for and situations.
Practice Questions
- 1 A fair coin is flipped and a fair six-sided die is rolled. What is the probability of getting heads and rolling a 4?
- 2 A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn without replacement. What is the probability that both marbles are red?
- 3 A student says that drawing a king from a deck and then drawing another king without replacement are independent events. Explain whether the student is correct and identify the correct multiplication rule to use.