With three events there are eight possible combinations of happening and not happening, from “all three” to “none”. When the events are independent – three separate machines, three free throws, three unrelated flights – each combination’s probability is a product of three factors, using P(X) for an event that happens and 1 − P(X) for one that does not.
The calculator adds the right combinations for you. “All three” and “none” are single products. “Exactly one” adds the three ways in which a single event occurs, and “exactly two” the three ways in which exactly one is missing. “At least one” is easiest through the complement: one minus the chance of none. The union P(A ∪ B ∪ C) from the inclusion–exclusion rule gives the same number, and is shown as a check.
Typical uses include reliability (will at least one of three backup systems work?), quality control (how likely is a batch to pass all three checks?) and games of chance. If the events are not independent, the products do not apply and you need the joint probabilities instead.
How to use the Probability of Three Events Calculator
- Probability of A – P(A)
- Probability of B – P(B)
- Probability of C – P(C)
Results update as you type, and everything is calculated in your browser.
How it works
Each of the eight outcomes is the product of P or 1 − P for the three events. The outputs add up the relevant outcomes. Inclusion–exclusion – P(A) + P(B) + P(C) − P(AB) − P(AC) − P(BC) + P(ABC) – confirms the “at least one” result.
Worked example
Three independent backup generators start with probabilities 50%, 40% and 30%. All three start: 0.5 × 0.4 × 0.3 = 6%. None starts: 0.5 × 0.6 × 0.7 = 21%, so at least one starts with probability 79%. Exactly one: 44%; exactly two: 0.14 + 0.09 + 0.06 = 29%; at least two: 35%.
The eight outcomes
| A | B | C | Probability (example) |
|---|---|---|---|
| ✓ | ✓ | ✓ | 6% |
| ✓ | ✓ | ✗ | 14% |
| ✓ | ✗ | ✓ | 9% |
| ✗ | ✓ | ✓ | 6% |
| ✓ | ✗ | ✗ | 21% |
| ✗ | ✓ | ✗ | 14% |
| ✗ | ✗ | ✓ | 9% |
| ✗ | ✗ | ✗ | 21% |
Inputs explained
| Field | What to enter |
|---|---|
| Probability of A – P(A) | In % |
| Probability of B – P(B) | In % |
| Probability of C – P(C) | In % |
What each result means
- All three occur – P(A ∩ B ∩ C) = [Probability of A – P(A)]×[Probability of B – P(B)]×[Probability of C – P(C)]÷10000
- At least one occurs – P(A ∪ B ∪ C) = 100 – (100 – [Probability of A – P(A)])×(100 – [Probability of B – P(B)])×(100 – [Probability of C – P(C)])÷10000
- Exactly one occurs = ([Probability of A – P(A)]×(100 – [Probability of B – P(B)])×(100 – [Probability of C – P(C)]) + (100 – [Probability of A – P(A)])×[Probability of B – P(B)]×(100 – [Probability of C – P(C)]) + (100 – [Probability of A – P(A)])×(100 – [Probability of B – P(B)])×[Probability of C – P(C)])÷10000
- Exactly two occur = ([Probability of A – P(A)]×[Probability of B – P(B)]×(100 – [Probability of C – P(C)]) + [Probability of A – P(A)]×(100 – [Probability of B – P(B)])×[Probability of C – P(C)] + (100 – [Probability of A – P(A)])×[Probability of B – P(B)]×[Probability of C – P(C)])÷10000
- At least two occur = [Exactly two occur] + [All three occur – P(A ∩ B ∩ C)]
- None occurs = (100 – [Probability of A – P(A)])×(100 – [Probability of B – P(B)])×(100 – [Probability of C – P(C)])÷10000
- Check: inclusion–exclusion union = [Probability of A – P(A)] + [Probability of B – P(B)] + [Probability of C – P(C)] – ([Probability of A – P(A)]×[Probability of B – P(B)] + [Probability of A – P(A)]×[Probability of C – P(C)] + [Probability of B – P(B)]×[Probability of C – P(C)])÷100 + [Probability of A – P(A)]×[Probability of B – P(B)]×[Probability of C – P(C)]÷10000
- P(A and B) = [Probability of A – P(A)]×[Probability of B – P(B)]÷100
- P(A and C) = [Probability of A – P(A)]×[Probability of C – P(C)]÷100
- P(B and C) = [Probability of B – P(B)]×[Probability of C – P(C)]÷100
Formula
P(all three) = P(A)P(B)P(C) · P(none) = (1 − P(A))(1 − P(B))(1 − P(C))
P(at least one) = 1 − P(none)
P(exactly one) = P(A)P(B′)P(C′) + P(A′)P(B)P(C′) + P(A′)P(B′)P(C)
Worked example
Inputs — Probability of A – P(A): 50 %; Probability of B – P(B): 40 %; Probability of C – P(C): 30 %.
Results — All three occur – P(A ∩ B ∩ C) = 6 %; At least one occurs – P(A ∪ B ∪ C) = 79 %; Exactly one occurs = 44 %; Exactly two occur = 29 %; At least two occur = 35 %; None occurs = 21 %; Check: inclusion–exclusion union = 79 %; P(A and B) = 20 %; P(A and C) = 15 %; P(B and C) = 12 %.
Key terms
- Union
- A ∪ B ∪ C, at least one of the events occurs.
- Intersection
- A ∩ B ∩ C, all of the events occur.
- Inclusion–exclusion principle
- A rule for the union that corrects for overlaps counted more than once.
Frequently asked questions
How do I find the probability that all three events happen?
For independent events multiply: 0.5 × 0.4 × 0.3 = 6%.
What is the probability that at least one of three events happens?
One minus the chance of none: 1 − 0.5 × 0.6 × 0.7 = 1 − 0.21 = 79%.
How do I calculate exactly one of three?
Add the three cases where one happens and two do not: 0.5×0.6×0.7 + 0.5×0.4×0.7 + 0.5×0.6×0.3 = 0.21 + 0.14 + 0.09 = 44%.
What is the inclusion–exclusion formula for three events?
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C).
Do the results add up to 100%?
Yes: none + exactly one + exactly two + all three = 21% + 44% + 29% + 6% = 100%.
What if the events are not independent?
You need the joint probabilities of each pair and of all three; the simple products no longer apply.