Probability Calculator

What the probability calculator does

This calculator combines probabilities in the ways that come up most often. Give it the chance of two events and it works out the odds of both happening, either happening, and neither. Switch to series mode and it handles one event repeated many times, telling you the chance it always happens, never happens, or happens at least once.

These are the building blocks of probability, the and, the or, and the opposite. Below is how each is worked out and the one assumption to keep in mind.

How to use it

1. Choose a mode, two events or a series of events.
2. Enter the probabilities as percentages: P(A) and P(B) for two events, or P(A) and the number of repetitions for a series.
3. Press Calculate for the combined probabilities, or Reset to clear it.

The two-event rules

For two events, the calculator reports several combinations. The intersection, written P(A and B), is the chance both happen, found by multiplying the two probabilities. The union, P(A or B), is the chance at least one happens, found by adding the two and subtracting the intersection so the overlap is not counted twice.

It also gives the complement of each event, the chance it does not happen, which is 100 percent minus its probability, and the symmetric difference, the chance exactly one of the two happens but not both. Together these cover almost every everyday question about two events.

The independence assumption

There is one important assumption behind the intersection. Multiplying the two probabilities to get the chance of both is only correct when the events are independent, meaning one happening does not change the odds of the other. Two coin flips are independent; the first result tells you nothing about the second.

When events are not independent, the multiplication no longer holds. The chance of drawing two aces from a deck without replacing the first is not simply one probability times the other, because removing the first ace changes what is left. So read these results as applying to independent events, and for events that influence each other, a conditional probability is the right tool instead.

A series of events

Series mode takes a single event and repeats it a set number of times, assuming each repetition is independent. It answers three questions. The chance the event happens every time is its probability multiplied by itself once per repetition. The chance it never happens is the chance of it not happening, multiplied the same way.

The most useful is often the chance it happens at least once, which is 100 percent minus the chance it never happens. This is the neat way to answer questions like the chance of at least one success in several tries, which is far easier to reach through its opposite than by adding up all the ways it could happen.

A worked example

Take two independent events each with a 50 percent chance, like two coin flips landing heads. The chance of both is 50 percent times 50 percent = 25 percent. The chance of at least one is 50 plus 50 minus 25 = 75 percent.

Now switch to a series: one event at 50 percent, repeated three times, like flipping heads three times. The chance it happens every time is 12.5 percent, the chance it never does is also 12.5 percent, and the chance of at least one is 87.5 percent.

Entering your values

Enter probabilities as percentages between 0 and 100. In two-event mode you supply P(A) and P(B); in series mode, P(A) and how many times it repeats. The results are shown as percentages. Remember that the combinations assume the events, or the repetitions, are independent of one another.

Questions people ask

How do I find the probability of both events?

For independent events, multiply their probabilities. That gives the intersection, the chance both happen.

How do I find the probability of either event?

Add the two probabilities and subtract the chance of both, so the overlap is not double counted. That gives the union, the chance at least one happens.

What does independence mean here?

That one event happening does not change the odds of the other. The multiplication rule for both events only holds when they are independent.

How do I find the chance of at least one success?

Work out the chance of no successes at all, then subtract from 100 percent. That is usually far simpler than adding up every way at least one could happen.

References

A quick note on where the methods here come from. The rules for combining probabilities, including independence and complements, 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