Lesson 68: Probability Multiplication Rule

New Concept

This rule calculates the probability of multiple independent events occurring. If events A and B are independent, the probability of both happening is found by multiplying their individual probabilities.

Multiplication Rule for Probability
> If events A and B are independent, then
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What’s next

This is just the foundation for compound probability. Next, you’ll master this rule by working through examples with dice, coins, and spinners to see it in action.

Property

If an experiment has two parts, with $m$ outcomes for the first and $n$ for the second, the total number of outcomes is $m \cdot n$.

Examples

• Flipping 3 coins: $2 \cdot 2 \cdot 2 = 8$ total outcomes.
• Choosing 1 of 5 main dishes and 1 of 3 desserts: $5 \cdot 3 = 15$ possible meals.

Explanation

Need to find all possible combos for a multi-step event? Forget drawing a giant tree diagram! Just multiply the number of choices for each step to get your total. It’s a super-fast shortcut for counting all your options.

Property

Two events are independent if the occurrence of one does not change the probability of the other. The outcome of one event does not influence the other at all.

Examples

• A coin landing on heads and a number cube rolling a 4.
• Drawing a card from a deck, putting it back, and drawing a second card.

Explanation

Think about rolling a die and a spinner landing on red. The die showing a 5 has absolutely no effect on what color the spinner lands on. The events are completely separate, like they are in their own little worlds.

Property

If events A and B are independent, then the probability of both happening is found by multiplying their individual probabilities: $P(\text{A and B}) = P(\text{A}) \cdot P(\text{B})$.

Examples

• $P(\text{heads and rolling a 6}) = P(\text{heads}) \cdot P(6) = \frac{1}{2} \cdot \frac{1}{6} = \frac{1}{12}$.
• $P(\text{red then blue marble, with replacement}) = \frac{2}{5} \cdot \frac{3}{5} = \frac{6}{25}$.

Explanation

Want to know the chances of two separate things both happening? Just multiply their probabilities together! This powerful rule lets you calculate the likelihood of a combined outcome in a snap, like finding a fraction of a fraction.