Probability Rules

Learn union and overlap, complement events.

Lesson 2 of 32 parts Practice Flashcards

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Union and Overlap

Objective: Explain union and overlap, apply the stated method, and check its conditions.

The union A∪B means A or B or both. Its probability is P(A)+P(B)−P(A∩B), because adding the event probabilities counts overlap twice. Mutually exclusive events have zero intersection; other pairs require the overlap adjustment. The word “or” in a union includes both events occurring.

Formulas

$P(A∪B)=P(A)+P(B)−P(A∩B)$

Worked examples

Try itIn 27 equally likely outcomes, A has 13, B has 14, and A∩B has 7. Find P(A∪B).Show answer

Step 1: Use inclusion-exclusion. Step 2: Union count=(13)+(14)−7=20. Step 3: P(A∪B)=20/27; overlap must be subtracted once.

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Union and Overlap

Objective: Explain union and overlap, apply the stated method, and check its conditions.

Formulas

$P(A∪B)=P(A)+P(B)−P(A∩B)$

Worked examples

Try itIn 27 equally likely outcomes, A has 13, B has 14, and A∩B has 7. Find P(A∪B).Show answer

Step 1: Use inclusion-exclusion. Step 2: Union count=(13)+(14)−7=20. Step 3: P(A∪B)=20/27; overlap must be subtracted once.

Complement Events

Objective: Explain complement events, apply the stated method, and check its conditions.

The complement Aᶜ contains all outcomes not in A. Since an event and its complement partition the sample space, P(Aᶜ)=1−P(A). The complement depends on the complete stated outcome space, including every possible alternative. Include every alternative in the stated sample space.

Formulas

$P(Aᶜ)=1−P(A)$

Worked examples

Try itA bag has 42 red and 49 blue counters. Find P(not red) for one uniform draw.Show answer

Step 1: Not red means blue in this two-color bag. Step 2: Use 49/(42+49). Step 3: The probability is 7/13.

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