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All courses Math II · S-CP.7 69 of 73
Apply and interpret the Addition Rule P(A or B)=P(A)+P(B)-P(A and B).

Use the Addition Rule to find P(A or B) for overlapping events

Problem
Use P(A or B)=P(A)+P(B)−P(A and B) with 0.50, 0.30, and overlap 0.10. Compute the union without double-counting the overlap.
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Hint

Use the addition rule for overlapping events: \(P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)\).

Subtract the overlap once because adding \(P(A)\) and \(P(B)\) counts the intersection twice.

Solution walkthrough

01

Use the overlapping-events addition rule

\[P(A~\text{or}~B)=P(A)+P(B)-P(A~\text{and}~B)\]

Adding both event probabilities counts overlap twice, so subtract it once.

02

Substitute

\[0.50+0.30-0.10\]

The source values are P(A)=0.50, P(B)=0.30, and joint probability 0.10.

03

Calculate

\[0.80-0.10=0.70\]

The inclusive union probability is 0.70.

04

Check bounds

\[0.50\le~0.70\le~0.80\]

The union is at least each marginal and no more than their uncorrected sum, a necessary consistency check.

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Another way

  1. Break the union into A-only 0.40, overlap 0.10, and B-only 0.20, then add.

!

Common mistake

Do not stop at 0.80; that double-counts the overlap.