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Probability and statistics
Addition Rule for probability
20 practice questions
2 video lessons
Theory + worked examples
Theory
The addition rule finds the probability that \(A\) or \(B\) occurs:
\[P(A\cup B)=P(A)+P(B)-P(A\cap B).\]
Subtract the overlap \(P(A\cap B)\) because it is counted in both \(P(A)\) and \(P(B)\). If the events are mutually exclusive the overlap is \(0\).
Add \(P(A)\) and \(P(B)\), then subtract the double-counted overlap.
The addition rule.
The addition rule:
\[P(A\cup B)=P(A)+P(B)-P(A\cap B)\]
Mutually exclusive events satisfy \(P(A\cap B)=0\), so \(P(A\cup B)=P(A)+P(B)\).
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 \(0\).
Example 1 — Overlapping events
Draw one card. Find \(P(\text{king or heart})\).
Solution
There are \(4\) kings, \(13\) hearts, and \(1\) card that is both.
| \(P\) | \(=\) | \(\dfrac{4}{52}+\dfrac{13}{52}-\dfrac{1}{52}\) |
| \(=\) | \(\dfrac{16}{52}=\dfrac{4}{13}\) |
Example 2 — Mutually exclusive
Roll a die. Find \(P(\text{a }2\text{ or a }5)\).
Solution
These cannot both happen, so just add.
| \(P\) | \(=\) | \(\dfrac16+\dfrac16=\dfrac26=\dfrac13\) |
Example 3 — Using the rule to find the overlap
\(P(A)=0.5,\ P(B)=0.4,\ P(A\cup B)=0.7\). Find \(P(A\cap B)\).
Solution
Rearrange the addition rule.
| \(P(A\cap B)\) | \(=\) | \(P(A)+P(B)-P(A\cup B)\) |
| \(=\) | \(0.5+0.4-0.7=0.2\) |
Example 4 — From a survey
\(60\%\) like tea, \(50\%\) like coffee, \(30\%\) like both. What percent like at least one?
Solution
Apply the addition rule.
| \(P\) | \(=\) | \(0.60+0.50-0.30\) |
| \(=\) | \(0.80=80\%\) |
Common pitfalls
Don't forget to subtract the overlap for events that can both occur.
Only skip the overlap when events are mutually exclusive.
“Or” adds, “and” multiplies — keep the rules distinct.
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.
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