Probability Calculator

How to use the probability calculator

1. Fill in the fields — they start with a worked example.
2. Read the answer, and the arithmetic shown beneath it.
3. Copy the results, or adjust the inputs to compare scenarios.

About this tool

Two rules cover most of what people need. Both events happen with probability P(A) × P(B) — but only when the events are independent, meaning neither changes the odds of the other. Either event happens with P(A) + P(B) − P(A and B), where the subtraction removes the overlap that would otherwise be counted twice.

Independence is the assumption that quietly breaks. Two servers failing, two flights delayed, two loans defaulting: these feel independent and usually are not, because they share a cause — the same data centre, the same weather, the same recession. Multiplying their probabilities then produces a combined risk that is far too low, which is a well-documented way that models understate correlated failure.

The repeated-trials figure is the one that surprises people. A one-in-a-hundred event has a sixty-three per cent chance of happening at least once in a hundred trials, not a certainty and not something negligible. That calculation runs through the complement — the chance of it never happening is 0.99 raised to the hundredth power — because working out the probability of nothing happening is much easier than summing every way something might.

Mutually exclusive events are offered as the other case: two outcomes that cannot both occur, where the probability of both is zero and the probability of either is simply the sum. Rolling a three and rolling a five on one die are mutually exclusive; they are not independent, and the two words are not interchangeable.

Common uses

• Working out the chance of at least one failure over many attempts.
• Combining two independent risks into one figure.
• Converting a per-attempt probability into an expected count.

Frequently asked questions

What is the probability of at least one occurrence?

One minus the probability of none. For a one-in-a-hundred event over a hundred trials that is 1 − 0.99¹⁰⁰, or about 63 per cent — not a certainty, and not negligible.

What is the difference between independent and mutually exclusive?

Independent events do not affect each other's odds, so both can happen. Mutually exclusive events cannot both happen at all. They are different ideas and often confused.

When does multiplying probabilities go wrong?

When the events share a cause. Two servers in one data centre, or two flights in one storm, are not independent, and multiplying their probabilities badly understates the combined risk.