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Probability Calculator

Know the chance of two things on their own? Find the chance that both, either, exactly one, or neither happens.

Level: Grades 10–11

  • Common Core standard HSS.CP.A.2Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
  • Common Core standard HSS.CP.B.7Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.
  • Common Core standard HSS.CP.A.1Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
Printable worksheet

Probability Calculator

Know the answer you need? Pick an input to work backward.

Inputs

A decimal from 0 to 1, for example 0.5 for a 50% chance.

A decimal from 0 to 1.

How the events relate

Result

P(A and B), both happen

0.15

With P(A) = 0.5 and P(B) = 0.3 (independent), both happen with probability 0.15 and at least one happens with probability 0.65.

P(A or B), at least one happens
0.65
P(not A)
0.5
P(not B)
0.7
P(A xor B), exactly one happens
0.5
P(neither)
0.35

See the step-by-step solution ↓

Step-by-step solution

  1. Multiply the two chances to get the chance both happen. This works because one event doesn't change the other's odds

    P(A and B)=P(A)×P(B)=0.5×0.3=0.15

  2. For at least one, add the two chances. Then take away the chance of both, so it isn't counted twice

    P(A or B)=P(A)+P(B)−P(A and B)=0.5+0.3−0.15=0.65

  3. For A not happening, take its chance away from 1, because an event either happens or it doesn't

    P(not A)=1−0.5=0.5

  4. For exactly one, start with at least one and take away the cases where both happen

    P(A xor B)=P(A or B)−P(A and B)=0.65−0.15=0.5

  5. For neither, take the chance of at least one away from 1. What's left is every case where nothing happens

    P(neither)=1−0.65=0.35

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find the chance that two things both happen, that at least one happens, or that neither does. Many everyday questions are like this. Will it rain on both days of the trip? Will at least one of two job applications succeed? Will exactly one of two machines break down this month? If you know the chance of each event on its own, and how the two relate, a few short rules give every answer.

Enter the probability (chance) of event A and of event B. You get six results at once: both happen, at least one happens, A doesn't happen, B doesn't happen, exactly one happens, and neither happens. You also choose how the events relate, because the rules differ. Independent means one tells you nothing about the other. Mutually exclusive means they can't both happen.

Where it's used

Why this math matters outside the classroom:

  • Weekend weather. With independent 30% and 40% chances of rain on Saturday and Sunday, the chance of a dry weekend is 0.7 × 0.6 = 42%.
  • Backup alarms. Two independent alarms that each work 95% of the time both fail only 0.05 × 0.05 = 0.25% of the time.
  • Card draws. Drawing a king or a queen from a deck, which can't both happen at once, has probability 4/52 + 4/52 = 8/52, about 15.4%.
  • At least one. A player who needs either of two independent 50% events has a 1 − 0.5 × 0.5 = 75% chance of getting at least one.
  • Quality tests. A part passing test A with probability 0.9 and test B with 0.8, independently, passes exactly one 0.9 × 0.2 + 0.1 × 0.8 = 26% of the time.

Grade level and standards: Grades 10–11; Common Core State Standards for Mathematics HSS.CP.A.2, HSS.CP.B.7, HSS.CP.A.1. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Enter the probability of event A as a decimal between 0 and 1. A 25% chance is 0.25.
  2. Enter the probability of event B the same way.
  3. Choose Independent if one event has no effect on the other. Choose Mutually exclusive if the two events can never happen together.
  4. Read the six probabilities and the working below them. To turn any result into a percent, multiply it by 100.

What the inputs mean

Probability of A, P(A)
The chance that event A happens, on a scale from 0 to 1. 0 means impossible and 1 means certain.
Probability of B, P(B)
The chance that event B happens, on the same scale.
How the events relate
Independent events don't affect each other, like two separate coin flips. Mutually exclusive events rule each other out, like rolling a 2 and rolling a 5 on a single die.

Formula and method

Independent:        P(A and B) = P(A) × P(B)
Mutually exclusive: P(A and B) = 0
P(A or B)  = P(A) + P(B) − P(A and B)
P(not A)   = 1 − P(A)
P(A xor B) = P(A or B) − P(A and B)
P(neither) = 1 − P(A or B)

In the formulas, P(A) means the probability of event A, and P(B) the probability of event B. "A and B" means both happen. "A or B" means at least one happens. "A xor B" means exactly one happens.

The multiplication rule works only for independent events. If knowing that A happened doesn't change the chance of B, the chance of both is the two chances multiplied. Mutually exclusive events never overlap, so the chance of both is zero.

The addition rule finds the chance that at least one event happens. Adding P(A) and P(B) counts the cases where both happen twice, so you subtract that overlap once. The complement rule says an event either happens or it doesn't, so those two chances add up to 1.

'Exactly one' is 'at least one' with the 'both' cases taken out. 'Neither' is everything left over after 'at least one', its complement.

Worked example

You flip a coin (A is heads, P(A) = 0.5). Separately, you spin a spinner with a 30% chance of landing on red (B is red, P(B) = 0.3). The two are independent.

  1. Both: 0.5 × 0.3 = 0.15.
  2. At least one: 0.5 + 0.3 − 0.15 = 0.65.
  3. Not A: 1 − 0.5 = 0.5, and not B: 1 − 0.3 = 0.7.
  4. Exactly one: 0.65 − 0.15 = 0.5.
  5. Neither: 1 − 0.65 = 0.35.

Result: There's a 15% chance of heads and red together, a 65% chance of at least one, a 50% chance of exactly one, and a 35% chance of neither. If the events were mutually exclusive instead, 'both' would be 0 and 'at least one' would be 0.8.

Understanding the result

A probability describes what happens over many tries. A result of 0.15 means that if you repeated the situation many times, both events would happen together about 15 times in every 100. It doesn't predict any single try.

The six results always fit together. 'Both', 'exactly one' and 'neither' cover every possibility without overlap, so they add up to 1. In the example, 0.15 + 0.5 + 0.35 = 1. That's a handy check that your inputs make sense.

Assumptions and limitations

  • In independent mode, the events truly don't affect each other. Drawing two cards from the same deck without putting the first one back is not independent.
  • In mutually exclusive mode, the events can't happen together, so P(A) + P(B) can't be more than 1. The calculator shows an error if it is.
  • You enter probabilities as decimals from 0 to 1, not as percents.
  • Some events are neither independent nor mutually exclusive, like rain today and rain tomorrow. Those need the conditional probability P(B given A): the chance of B once you know A happened. This calculator doesn't take that as an input.
  • It handles only two events. For several independent events, multiply all their probabilities to get the chance that all of them happen.
  • The results are only as good as the probabilities you enter. Estimates based on small samples carry their own uncertainty.

Frequently asked questions

What is the difference between independent and mutually exclusive events?

Independent events don't affect each other. Both can happen, and the chance of both is their probabilities multiplied. Mutually exclusive events can't happen together at all, so the chance of both is zero. Two events with chances above zero can't be independent and mutually exclusive at the same time. If one happening rules out the other, they clearly affect each other.

Why do I subtract P(A and B) when finding P(A or B)?

Adding P(A) and P(B) counts the outcomes where both happen twice: once inside A and once inside B. Subtracting the overlap once fixes the double count. When events are mutually exclusive, the overlap is zero, so P(A or B) is just P(A) + P(B).

How do I enter a percentage or a fraction?

Change it to a decimal first. Divide a percent by 100, so 40% becomes 0.4. Divide the top of a fraction by the bottom, so 1/6 becomes about 0.1666667. Typing more decimal places gives a more precise result.

What does 'A xor B' mean?

It means exactly one of the two events happens, but not both. The name comes from 'exclusive or' in logic. It equals P(A) + P(B) − 2 × P(A and B).

Can I use this for more than two events?

Not directly. For independent events, the chance that all of them happen is all their probabilities multiplied together. The chance that none happen is their complements (1 minus each chance) multiplied together. The chance of at least one is 1 minus that last number.

How this calculator is checked

Method: Standard multiplication, addition and complement rules of probability for two events. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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