Probability of 3 Events Calculator
Enter the probability of each of three independent events as a percentage and get every useful combination instantly: all three together, at least one, exactly one, at least two, exactly two, and none at all. The "show your work" panel traces each formula with your actual numbers so you can verify the arithmetic or use it in an assignment.
What this calculator computes
When you have three separate events, each with its own probability, you rarely care about just one combination. You might want the chance all three succeed, or the chance at least one succeeds, or the chance exactly two succeed out of three. This calculator computes all six standard combinations at once: all three, at least one, exactly one, at least two, exactly two, and none. You enter the probability of each event as a percentage and every combination updates instantly. The "show your work" panel traces every formula with your actual numbers so the arithmetic is transparent.
The formulas and how they work
All formulas assume the three events are independent, meaning the outcome of one has no effect on the others. The joint probability (all three) is simply the product P(A) * P(B) * P(C). The union (at least one) requires the inclusion-exclusion principle: add the three individual probabilities, subtract the three pairwise products (because each pair is counted twice), then add back the triple product (because it was subtracted one too many times). Exactly one is the sum of three scenarios where one event happens and the other two do not: for each scenario multiply the one event probability by the complements of the other two. Exactly two is the same idea applied to pairs: for each pair multiply both probabilities by the complement of the third, then sum all three pairs. At least two is exactly two plus all three. None is simply 1 minus the union probability.
Independence, dependence, and when this tool applies
Two events are independent when knowing one outcome gives no information about the other. Rolling a die twice, flipping coins, drawing from separate decks: all independent. Many real-world events are not independent. If event A is "it rains today" and event B is "the match is cancelled," A and B are correlated because rain causes cancellations. When events are dependent you need conditional probabilities such as P(B | A), and the multiplication rule becomes P(A and B) = P(A) * P(B | A) rather than the simpler product. This calculator implements the independent version, which is exact for truly independent events and a reasonable approximation when correlation is weak.
Worked example: quality control
Suppose a production line has three inspection stages. Stage A passes 90% of items, stage B passes 80%, and stage C passes 70%. Enter 90, 80 and 70 to get: all three pass = 50.40%; at least one passes = 99.40%; exactly one passes = 14.96%; exactly two pass = 34.04%; none pass = 0.60%. The "all three pass" figure (50.40%) is the overall yield of the production line if items must pass every stage. The "at least one passes" figure tells you how often an item clears at least one hurdle, which might matter if partial completion has value.
Probability combination formulas for three independent events
| Combination | Formula | Notes |
|---|---|---|
| All three - P(A and B and C) | P(A) * P(B) * P(C) | Multiplication rule for independent events |
| At least one - P(A or B or C) | P(A)+P(B)+P(C)-P(AB)-P(AC)-P(BC)+P(ABC) | Inclusion-exclusion principle |
| Exactly one | P(A)*P(B')*P(C') + P(A')*P(B)*P(C') + P(A')*P(B')*P(C) | Sum of three mutually exclusive scenarios |
| Exactly two | P(A)*P(B)*P(C') + P(A)*P(B')*P(C) + P(A')*P(B)*P(C) | Sum of pairwise intersections excluding the third |
| At least two | P(exactly two) + P(all three) | Shortcut using other computed values |
| None | 1 - P(at least one) | Complement rule applied to the union |
All formulas assume events A, B and C are mutually independent.
Frequently asked questions
What does "at least one" mean?
"At least one" means one, two, or all three events occur. It is the union of the three events and is calculated using the inclusion-exclusion formula: P(A) + P(B) + P(C) - P(AB) - P(AC) - P(BC) + P(ABC). The complement of "at least one" is "none of the three occur," so you can also get it as 1 minus P(none).
Why do I subtract pairwise probabilities in the union formula?
When you add P(A) + P(B) + P(C), every overlap is counted more than once. The pair P(A and B) is included in both P(A) and P(B), so you subtract it once. The same applies to P(A and C) and P(B and C). But then the triple overlap P(A and B and C) has been subtracted three times from the pairwise step, so you add it back once. This systematic correction is the inclusion-exclusion principle.
What is the difference between exactly two and at least two?
"Exactly two" means precisely two of the three events happen and the third does not. "At least two" means two or three of the events happen. The relationship is: P(at least two) = P(exactly two) + P(all three). When the individual probabilities are small, these two values are close; when probabilities are high, P(all three) becomes significant and the gap widens.
Can I use this for dependent events?
No. All formulas here assume independence: P(A and B) = P(A) * P(B). For dependent events you need the general multiplication rule P(A and B) = P(A) * P(B | A), where P(B | A) is the conditional probability of B given A occurred. If you know the conditional probabilities you can compute the intersections separately and then plug them into the inclusion-exclusion formula for the union.
What if the probabilities are given as fractions or decimals instead of percentages?
Convert to percentages by multiplying by 100. A probability of 0.75 becomes 75%, and a probability of 3/4 is 75%. All six output probabilities are displayed as percentages as well, so you can directly compare them. If you need the raw decimal, divide the displayed percentage by 100.
Do the six output probabilities add up to 100%?
No, they are not mutually exclusive so they cannot sum to 100%. For example, "all three occur" is already included inside "at least one occurs." The four mutually exclusive and exhaustive cases are: none, exactly one, exactly two, and all three. Those four do sum to 100%, and you can verify this with the calculator by adding those four outputs together.