Probability of 3 Events Calculator
Enter the probability of three independent events to find P(all), P(at least one), P(none), and P(A union B union C).
What is the Probability of 3 Events?
When three separate events A, B, and C can each independently occur or not occur, you often need to answer four distinct questions: What is the probability that all three happen? What is the probability that at least one happens? What is the probability that none happen? And what is the probability that A, B, or C (or some combination) happen, expressed as the union?
For independent events — meaning the occurrence of one event has no effect on the probability of the others — the mathematics is elegant and systematic. The probability that all three events occur simultaneously is simply the product of their individual probabilities: P(A) × P(B) × P(C). This is the multiplication rule for independent events extended from two events to three.
The probability that at least one event occurs is best computed using the complement rule: instead of laboriously adding all the ways at least one can happen (exactly one, exactly two, or all three), you compute P(none) and subtract from 1. P(none) = (1−P(A)) × (1−P(B)) × (1−P(C)), then P(at least one) = 1 − P(none). This is one of the most powerful shortcuts in probability theory.
The union P(A ∪ B ∪ C) uses the inclusion-exclusion principle, one of the cornerstones of combinatorics and probability. You start by adding all three individual probabilities, subtract each pairwise intersection (which you have double-counted), then add back the triple intersection (which you have subtracted once too many). The formula is: P(A) + P(B) + P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C).
Real-world applications of three-event probability are widespread. Risk analysts compute the probability that three independent systems all fail (reliability engineering). Medical researchers ask: given a 70% chance of test A detecting disease, 60% for test B, and 80% for test C, what is the probability at least one test detects it? Sports analysts estimate the probability that three different athletes all perform above their expected level in the same game. Financial modelers assess the joint probability of three market conditions occurring simultaneously.
Formulas
All three events (joint probability):
None of the three events:
At least one event occurs:
Union (A or B or C or any combination):
Note: For independent events, P(A∩B) = P(A)×P(B), P(A∩C) = P(A)×P(C), P(B∩C) = P(B)×P(C), and P(A∩B∩C) = P(A)×P(B)×P(C).
How to Use This Calculator
Steps to Calculate
Example Calculations
Example 1 — General Three Events
P(A) = 60%, P(B) = 40%, P(C) = 30%
Example 2 — Equal Probabilities
P(A) = 50%, P(B) = 50%, P(C) = 50%
Example 3 — High Probability Events
P(A) = 80%, P(B) = 70%, P(C) = 60%
Example 4 — Rare Events (Risk Assessment)
P(A) = 10%, P(B) = 20%, P(C) = 30%
Frequently Asked Questions
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How do you find the probability that all 3 independent events occur?
For independent events, P(A and B and C) = P(A) × P(B) × P(C). Convert percentages to decimals first. For example, P(A)=60%, P(B)=40%, P(C)=30%: P(all) = 0.60 × 0.40 × 0.30 = 0.072 = 7.2%.
What is the formula for P(A union B union C)?
P(A∪B∪C) = P(A) + P(B) + P(C) - P(A∩B) - P(A∩C) - P(B∩C) + P(A∩B∩C). For independent events, each intersection is the product of the individual probabilities.
How do you calculate the probability that at least one of 3 events occurs?
P(at least one) = 1 - P(none of them occur) = 1 - (1-P(A))(1-P(B))(1-P(C)). This complement approach is far simpler than adding up all cases where exactly one, exactly two, or all three occur.
What is the probability that none of 3 events occur?
For independent events: P(none) = (1-P(A)) × (1-P(B)) × (1-P(C)). This is the joint probability that A fails AND B fails AND C fails simultaneously.
Are these events independent? What changes if they are not?
This calculator assumes all three events are independent. If events are dependent (correlated), you cannot simply multiply probabilities. Instead you need conditional probabilities: P(A∩B∩C) = P(A) × P(B|A) × P(C|A∩B). Dependent event calculations require more information about the conditional relationships.
P(A)=0.5, P(B)=0.4, P(C)=0.3 - what is P(all three)?
P(all three) = 0.5 × 0.4 × 0.3 = 0.060 = 6.0%. As percentages: 50% × 40% × 30% / 10000 = 6.0%. The at-least-one probability = 1 - (0.5)(0.6)(0.7) = 1 - 0.21 = 0.79 = 79%.
What is the inclusion-exclusion principle for 3 events?
The inclusion-exclusion principle states: |A∪B∪C| = |A| + |B| + |C| - |A∩B| - |A∩C| - |B∩C| + |A∩B∩C|. For probabilities: P(A∪B∪C) = P(A)+P(B)+P(C) - P(A∩B) - P(A∩C) - P(B∩C) + P(A∩B∩C). You first add individual probabilities, subtract the pairwise overlaps, then add back the triple overlap you subtracted too many times.
How is P(at least one) related to P(none)?
P(at least one) + P(none) = 1. They are complements. If P(none) = 0.20, then P(at least one) = 0.80. This complement relationship makes calculating P(at least one) much simpler: compute P(none) as a product of individual complement probabilities, then subtract from 1.
What if two events are mutually exclusive?
If two events are mutually exclusive (cannot both occur), their joint probability is 0. For example, if A and B are mutually exclusive, P(A∩B) = 0, and P(A∩B∩C) = 0. Mutually exclusive events are always dependent. This calculator assumes no mutual exclusivity.
How do I verify my 3-event probability calculation?
Check: P(all three) should be the smallest value, P(at least one) the largest. P(at least one) + P(none) must equal 100%. P(union) = P(at least one) for independent events. Also verify P(all three) ≤ each individual probability. If P(A)=60%, P(all three) cannot exceed 60%.