Algebra 2
Statistics and probability
Probability rules (addition, multiplication)
20 practice questions
0 video lessons
Theory + worked examples
Probability Rules
California Algebra 2 • Standard S-CP.7 • Statistics & Probability
Probability Rules is a topic in Statistics & Probability in the California Common Core State Standards. It is aligned to Standard S-CP.7, which requires students to apply the Addition Rule and the general Multiplication Rule and interpret the answers in context.
The addition rule gives \(P(A\cup B)\) by subtracting the overlap; the multiplication rule gives \(P(A\cap B)\) as a product.
Theory
Two rules combine probabilities:
- Addition: \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\).
- Multiplication: \(P(A\cap B)=P(A)\,P(B\mid A)\).
- For independent events, \(P(A\cap B)=P(A)\,P(B)\).
“Or” adds; “and” multiplies — and subtract any overlap.
The addition rule subtracts the overlap \(A\cap B\).
The probability rules.
Addition and multiplication:
\[P(A\cup B)=P(A)+P(B)-P(A\cap B),\quad P(A\cap B)=P(A)\,P(B\mid A)\]
Mutually exclusive: the overlap is \(0\). Independent: \(P(B\mid A)=P(B)\).
How to combine probabilities
- For “or”, add and subtract the overlap.
- For “and”, multiply.
- Use \(P(B\mid A)\) if the events are dependent.
- Check for mutually exclusive or independent to simplify.
Example 1 — Addition rule
Draw one card. Find \(P(\text{king or heart})\).
Solution
Subtract the overlap (king of hearts).
| \(\dfrac{4}{52}+\dfrac{13}{52}-\dfrac{1}{52}\) | ||
| \(=\) | \(\dfrac{16}{52}=\dfrac{4}{13}\) |
Example 2 — Multiplication (independent)
Flip two coins. Find \(P(\text{two heads})\).
Solution
Independent events multiply.
| \(\dfrac12\cdot\dfrac12\) | \(=\) | \(\dfrac14\) |
Example 3 — Mutually exclusive
Roll a die. Find \(P(2\text{ or }5)\).
Solution
These can't both happen, so no overlap.
| \(\dfrac16+\dfrac16\) | \(=\) | \(\dfrac13\) |
Example 4 — Dependent multiplication
Draw two cards without replacement. Find \(P(\text{two kings})\).
Solution
The second draw is conditional.
| \(\dfrac{4}{52}\cdot\dfrac{3}{51}\) | \(=\) | \(\dfrac{1}{221}\) |
Common pitfalls
Subtract the overlap in the addition rule unless mutually exclusive.
Use \(P(B\mid A)\) when the second event depends on the first.
Independent \(\neq\) mutually exclusive.
Frequently asked questions
What is the addition rule?
\(P(A\cup B)=P(A)+P(B)-P(A\cap B)\).
What is the multiplication rule?
\(P(A\cap B)=P(A)P(B\mid A)\).
What does mutually exclusive mean?
The events can't both happen, so the overlap is \(0\).
When do you just multiply P(A) and P(B)?
When the events are independent.
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Conditional probability and independence
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