Union Probability Calculator

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Calculate the probability that event A OR event B occurs using the inclusion-exclusion principle.

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What is Union Probability?

The union of events A and B, written A ∪ B, is the event that A occurs, OR B occurs, OR both occur. We say "A union B" or "A or B".

The Inclusion-Exclusion Formula

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

P(A ∪ B) = Probability that A or B (or both) occurs

P(A) = Probability of event A

P(B) = Probability of event B

P(A ∩ B) = Probability that both A and B occur (intersection)

Why Subtract P(A ∩ B)?

When we add P(A) + P(B), outcomes in the overlap get counted twice. We subtract P(A ∩ B) to correct for this double-counting.

Visualizing with Venn Diagrams

A only

P(A) - P(A ∩ B)

A ∩ B

Overlap

B only

P(B) - P(A ∩ B)

Special Case: Mutually Exclusive Events

Events are mutually exclusive when they cannot occur at the same time. For these events, P(A ∩ B) = 0.

Simplified formula for mutually exclusive events:

P(A ∪ B) = P(A) + P(B)

Examples of mutually exclusive events:

Rolling a 1 OR rolling a 6 on a die
Drawing a red marble OR drawing a blue marble (single draw)
Getting heads OR getting tails on a single coin flip

Example: Even OR Greater Than 4

Problem

Roll a fair die. What is P(Even OR Greater than 4)?

Find P(A) - Even numbers

Even = {2, 4, 6}, so P(Even) = 3/6 = 1/2

Find P(B) - Greater than 4

GT4 = {5, 6}, so P(GT4) = 2/6 = 1/3

Find P(A ∩ B) - Both even AND GT4

Both = {6}, so P(Both) = 1/6

Apply inclusion-exclusion

Answer

P(Even OR Greater than 4) = 2/3 or about 66.67%.

Verification: {2, 4, 5, 6} = 4 outcomes out of 6 = 2/3

Quick Reference

General Union Formula

Mutually Exclusive Events

Union of Three Events

Frequently Asked Questions

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What is Union Probability?

The union of events A and B, written A ∪ B, is the event that A occurs, OR B occurs, OR both occur. We say "A union B" or "A or B".

The Inclusion-Exclusion Formula

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

P(A ∪ B) = Probability that A or B (or both) occurs

P(A) = Probability of event A

P(B) = Probability of event B

P(A ∩ B) = Probability that both A and B occur (intersection)

Why Subtract P(A ∩ B)?

When we add P(A) + P(B), outcomes in the overlap get counted twice. We subtract P(A ∩ B) to correct for this double-counting.

Special Case: Mutually Exclusive Events

Events are mutually exclusive when they cannot occur at the same time. For these events, P(A ∩ B) = 0.

Simplified formula for mutually exclusive events:

P(A ∪ B) = P(A) + P(B)

Examples of mutually exclusive events:

Rolling a 1 OR rolling a 6 on a die
Drawing a red marble OR drawing a blue marble (single draw)
Getting heads OR getting tails on a single coin flip

Example: Even OR Greater Than 4

Problem

Roll a fair die. What is P(Even OR Greater than 4)?

Find P(A) - Even numbers

Even = {2, 4, 6}, so P(Even) = 3/6 = 1/2

Find P(B) - Greater than 4

GT4 = {5, 6}, so P(GT4) = 2/6 = 1/3

Find P(A ∩ B) - Both even AND GT4

Both = {6}, so P(Both) = 1/6

Apply inclusion-exclusion

Answer

P(Even OR Greater than 4) = 2/3 or about 66.67%.

Verification: {2, 4, 5, 6} = 4 outcomes out of 6 = 2/3