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Glacius
ProbabilityConcept reference

Union probabilities

The probability of a two-event union is the sum of the event probabilities minus their intersection probability.

1 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. The union A ∪ B means A or B, including both
  3. For a uniformly selected record, the same correction applies to counts
  4. Check your understanding
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01The union A ∪ B means A or B, including both#

The union A ∪ B means A or B, including both. An outcome in the overlap belongs to the union just once.

Here A has mass 0.5 and B has 0.4. Adding counts the shared 0.3 twice, so subtract one copy: 0.9 − 0.3 = 0.6.

A and B overlap. The left-only region has mass 0.2, the shared region 0.3, and the right-only region 0.1. A has mass 0.5 and B has mass 0.4; their union has mass 0.6. Region sizes are schematic, so printed numbers carry the probabilities.A and B overlap. The left-only region has mass 0.2, the shared region 0.3, and the right-only region 0.1. A has mass 0.5 and B has mass 0.4; their union has mass 0.6. Region sizes are schematic, so printed numbers carry the probabilities.
Figure 1A and B overlap. The left-only region has mass 0.2, the shared region 0.3, and the right-only region 0.1. A has mass 0.5 and B has mass 0.4; their union has mass 0.6. Region sizes are schematic, so printed numbers carry the probabilities.
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For the probability of A∪BA\cup B:

P(A)+P(B)−P(A∩B)\begin{gathered}P(A)+P(B)\\{}-P(A\cap B)\end{gathered}

Subtract only the overlap that was counted twice. With no overlap, there is nothing to subtract.

Check your reasoning

P(A)=0.5, P(B)=0.4, P(A∩B)=0.2. Find P(A∪B).

Show answer and explanation
0.7

0.5 + 0.4 − 0.2 = 0.7.

02For a uniformly selected record, the same correction applies to counts#

For a uniformly selected record, the same correction applies to counts. With 12 in A, 9 in B and 3 in both, 18 distinct records belong to either event.

Check your reasoning

Choose one of 40 records uniformly. 16 have A, 12 have B, and 4 have both. Find P(A or B), including both.

Show answer and explanation
0.6

24 distinct records qualify; divide by 40.

03Check your understanding#

Check your reasoning

P(A)=0.6, P(B)=0.5, P(A∩B)=0.2. A learner gives P(A∪B)=0.7. Repair it.

  1. A0.7
  2. B1.1
  3. C0.9
Show answer and explanation
0.9

0.6 + 0.5 − 0.2 = 0.9.

Key takeaway

Add the two event probabilities, then subtract their shared mass once. The union includes outcomes in both events.

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Union probabilities.” Math behind ML. /learn/p-union