Probability Of 3 Events Calculator

What the probability of 3 events calculator does

This calculator extends the usual probability rules to three events at once. From the chances of events A, B, and C, it works out four combined results: the chance all three happen, the chance at least one does, the chance exactly one does, and the chance none of them do.

Three events bring more overlaps than two, which makes the arithmetic fiddlier by hand. Below is how each result is found and the assumption behind them.

How to use it

1. Enter P(A), P(B), and P(C), choosing decimals or percent for each with the selector beside it.
2. Press Calculate for the four combined probabilities, or Reset to clear it.

How the four results are worked out

The chance all three happen is the three probabilities multiplied together. The chance of none is the chance of at least one, subtracted from certainty. The chance of exactly one adds up the three ways a single event can happen while the other two do not.

The chance of at least one is the trickiest, because the events can overlap, and that is where a special rule comes in. Each result rests on combining the individual probabilities in a way that respects how the events can and cannot happen together.

The overlap problem, and how it is solved

You might think the chance of at least one is just the three probabilities added together, but that overcounts. Where two events overlap, that overlap gets counted in both, and where all three overlap, it gets counted three times. Simply adding would push the total too high, sometimes past 100 percent, which is impossible.

The fix is a rule called inclusion-exclusion. You add the three single probabilities, subtract the three pairwise overlaps to correct the double counting, then add back the triple overlap that the subtraction removed once too often. The result is the true chance of at least one, with every region of the overlap counted exactly once.

The independence assumption

The calculator treats the three events as independent, meaning none of them changes the odds of the others. That is what lets it find each overlap by multiplying the relevant probabilities together, and it is the right assumption for things like separate dice, separate coin flips, or unrelated events.

If your events influence one another, the multiplied overlaps no longer hold, and the results will not be accurate. So these figures apply when the events are genuinely independent, which covers most of the everyday questions people bring to three events.

A worked example

Suppose you roll three dice and ask about getting a six on each, so P(A), P(B), and P(C) are each one in six, about 16.67 percent.

The chance all three are sixes is one in 216, about 0.46 percent. The chance of at least one six is about 42.13 percent, found by inclusion-exclusion, and its opposite, the chance of no sixes at all, is about 57.87 percent. The chance of exactly one six is about 34.72 percent. Notice how the chance of at least one, near 42 percent, is well below the 50 percent you would wrongly get by simply adding three sixteen-percents.

Entering your values

Enter each probability and pick decimals or percent to match how you typed it, so 0.5 as a decimal or 50 as a percent. The four results are shown as percentages. As with all these combinations, they assume the three events are independent of one another.

Questions people ask

How do I find the chance all three happen?

For independent events, multiply the three probabilities together. That gives the chance of all three at once.

Why can't I just add the probabilities for at least one?

Because adding overcounts the overlaps where events happen together. Inclusion-exclusion corrects this by subtracting the pairwise overlaps and adding back the triple overlap.

How is the chance of none found?

By subtracting the chance of at least one from certainty. If at least one has a 42 percent chance, then none has the remaining 58 percent.

Does this assume the events are independent?

Yes. The overlaps are found by multiplying probabilities, which only holds for independent events. For events that affect each other, these results will not be accurate.

References

A quick note on where the methods here come from. The rules for combining several events, including inclusion-exclusion, are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering probability.

1. NIST/SEMATECH e-Handbook of Statistical Methods (probability). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (probability topics). https://openstax.org/details/books/introductory-statistics-2e