How this instrument works
When three events are independent — meaning the outcome of one has no bearing on the others — the probability that all three happen together is simply the product of their individual probabilities: P(A and B and C) = P(A) × P(B) × P(C). This is the multiplication rule extended from two events to three, and it works because independence means you can multiply probabilities together without worrying about any one event shifting the odds of the others.
The result is always smaller than the smallest of the three individual probabilities, as long as each one is below 100% — exactly what you'd expect, since piling on more required conditions can only make the combined outcome less likely, never more. A 50% chance of A, 50% of B, and 50% of C combine to a 12.5% chance of all three, a useful reminder of how quickly the odds of a compound event shrink as more independent conditions stack on top of each other.
This calculator is specifically for independent events — flipping three separate coins, rolling three separate dice, three unrelated system failures. If the events are not independent (say, drawing three cards from a deck without replacement, where each draw changes the odds for the next), simple multiplication of the raw probabilities gives the wrong answer, and a conditional-probability approach is needed instead.
- Enter the probability of the first event into P(A), as a percentage.
- Enter the probability of the second event into P(B), as a percentage.
- Enter the probability of the third event into P(C), as a percentage.
- Read P(A and B and C) — the chance all three independent events occur together.
Worked example — three fair coins landing heads
You flip three fair coins. Each has a 50% chance of landing heads, and the coins don't influence each other, so the three flips are independent. Enter 50 into P(A), 50 into P(B), and 50 into P(C).
The calculator returns P(A and B and C) = 0.50 × 0.50 × 0.50 × 100 = 12.5%. You can check this by listing all 8 equally likely outcomes of three coin flips (HHH, HHT, HTH, THH, HTT, THT, TTH, TTT) — exactly 1 of the 8, HHH, has all three landing heads, and 1/8 = 12.5%, confirming the multiplication rule.
Questions
Does this formula work for dependent events too?
No — straight multiplication of P(A), P(B), and P(C) only gives the correct combined probability when the three events are independent. If one event changes the odds of another (for example, drawing cards from a deck without putting them back), you need each event's conditional probability given the ones before it: P(A and B and C) = P(A) × P(B given A) × P(C given A and B). Using this calculator's simple multiplication on dependent events will overstate or understate the true combined probability.
Why is the combined probability always smaller than each one alone?
Because you're multiplying numbers that are each 100% or less, and multiplying fractions or percentages under 100% together always produces a result no larger than the smallest one you started with. Requiring three separate things to all happen is a stricter condition than requiring just one of them, so the odds can only go down — never up — as more independent requirements are added to the list.
What if one of the probabilities is 100%?
If any one event is certain (100%), it drops out of the multiplication entirely — P(A and B and C) simply becomes the product of the other two, since multiplying by 1.0 changes nothing. This makes sense intuitively: a guaranteed event doesn't add any additional chance for the combination to fail, so the combined probability is governed entirely by whichever events are still uncertain.
How is this different from P(A or B or C)?
P(A and B and C) asks for the probability that every one of the three events happens, which is what this calculator computes by multiplying. P(A or B or C) asks for the probability that at least one of them happens, which is a larger number computed differently — typically via the inclusion-exclusion principle, since it must avoid double-counting overlapping outcomes. The two questions sound similar but have very different answers, especially as the number of events grows.