Pre-Algebra
Probability (introductory)
Probability of compound events
20 practice questions
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Theory + worked examples
Probability of Compound Events
California Pre-Algebra • Standard 7.SP.8 • Probability
Probability of Compound Events is a topic in Probability in the California Common Core State Standards. It is aligned to Standard 7.SP.8, which requires students to find the probability of compound events.
A compound event combines two or more simple events; for independent events, multiply the probabilities.
Theory
A compound event involves two or more simple events.
Independent events: \(P(A\text{ and }B)=P(A)\cdot P(B)\); otherwise count the full sample space.
The two-dice sample space.
Compound events.
Independent 'and':
\[P(A\text{ and }B)=P(A)\cdot P(B)\]
List or count the sample space when unsure.
How to find a compound probability
- Decide if the events are independent.
- Independent 'and': multiply.
- Otherwise, count favorable outcomes in the sample space.
- Divide by the total outcomes.
Example 1 — Two coins
Find \(P(\text{two heads})\) on two coins.
Solution
Multiply independent probabilities.
| \(\dfrac{1}{2}\cdot\dfrac{1}{2}\) | \(=\) | \(\dfrac{1}{4}\) |
Example 2 — Two dice sum
Find \(P(\text{sum}=7)\) with two dice.
Solution
Six favorable of \(36\).
| \(\dfrac{6}{36}\) | \(=\) | \(\dfrac{1}{6}\) |
Example 3 — And (independent)
\(P(\text{head})\) then \(P(\text{roll }6)\)?
Solution
Multiply.
| \(\dfrac{1}{2}\cdot\dfrac{1}{6}\) | \(=\) | \(\dfrac{1}{12}\) |
Example 4 — Count
How many outcomes for two dice?
Solution
\(6\times6\).
| \(6\cdot6\) | \(=\) | \(36\) |
Common pitfalls
Multiply only for independent 'and' events.
Count the full sample space when needed.
Two dice have \(36\) outcomes, not \(12\).
Frequently asked questions
What is a compound event?
An event combining two or more simple events.
How do I find P(A and B) for independent events?
Multiply \(P(A)\) and \(P(B)\).
How many outcomes for two dice?
\(36\).
\(P(\text{sum}=7)\)?
\(\dfrac16\).
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