Addition Rule for probability
The Addition Rule
The Addition Rule is a topic in Probability & Statistics in the California Common Core State Standards. It is aligned to Standard S-CP.7, which requires students to apply the Addition Rule and interpret it in terms of the model.
The addition rule gives \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\), subtracting the double-counted overlap.
Theory
The addition rule finds the probability that \(A\) or \(B\) occurs:
The addition rule:
How to use the addition rule
- Find \(P(A)\) and \(P(B)\).
- Find the overlap \(P(A\cap B)\).
- Add and subtract: \(P(A)+P(B)-P(A\cap B)\).
- If mutually exclusive, the overlap is \(0\).
There are \(4\) kings, \(13\) hearts, and \(1\) card that is both.
| \(P\) | \(=\) | \(\dfrac{4}{52}+\dfrac{13}{52}-\dfrac{1}{52}\) |
| \(=\) | \(\dfrac{16}{52}=\dfrac{4}{13}\) |
These cannot both happen, so just add.
| \(P\) | \(=\) | \(\dfrac16+\dfrac16=\dfrac26=\dfrac13\) |
Rearrange the addition rule.
| \(P(A\cap B)\) | \(=\) | \(P(A)+P(B)-P(A\cup B)\) |
| \(=\) | \(0.5+0.4-0.7=0.2\) |
Apply the addition rule.
| \(P\) | \(=\) | \(0.60+0.50-0.30\) |
| \(=\) | \(0.80=80\%\) |
Common pitfalls
Frequently asked questions
What is the addition rule for probability?
\(P(A\cup B)=P(A)+P(B)-P(A\cap B)\).
Why do you subtract the overlap?
Because outcomes in both events are counted twice when you add \(P(A)\) and \(P(B)\).
What are mutually exclusive events?
Events that cannot both occur, so \(P(A\cap B)=0\).
What is the addition rule for mutually exclusive events?
\(P(A\cup B)=P(A)+P(B)\), since the overlap is zero.