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Probability Calculator

Enter the probabilities of two independent events, A and B, to see every way they can combine: both happening, at least one, exactly one, or neither. The bar shows how the four possible outcomes divide up 100%.

Quick examples
%
%

Enter percentages: 0.25 as a decimal is 25%.

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Both A and B15%
A or B (at least one)
65%
Exactly one of them
50%
Neither
35%
Not A
50%
Not B
70%
A but not B
35%
B but not A
15%
How the four outcomes split
  • Both
  • Only A
  • Only B
  • Neither

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      Formula

      P(A and B) = P(A) × P(B)
      P(A or B) = P(A) + P(B) − P(A) × P(B)
      P(exactly one) = P(A) × (1 − P(B)) + P(B) × (1 − P(A))
      P(neither) = (1 − P(A)) × (1 − P(B))

      How to use it

      1. Enter the probability of event A as a percentage.
      2. Enter the probability of event B.
      3. Read the combined probabilities.

      Worked examples

      A has a 50% chance and B a 30% chance

      Both A and B
      15%
      A or B (at least one)
      65%
      Exactly one of them
      50%
      Neither
      35%

      A 40% chance of rain and a 20% chance of a delay

      Both A and B
      8%
      A or B (at least one)
      52%
      Exactly one of them
      44%
      Neither
      48%
      A but not B
      32%
      B but not A
      12%

      Independent events only

      These formulas assume that A happening does not change the chance of B — two coin flips, two separate machines, rain today and a train delay tomorrow. If the events affect each other, P(A and B) is P(A) times the probability of B given A, and the simple product no longer applies.

      Events that cannot both happen (drawing a card that is both a heart and a spade) are mutually exclusive, not independent: for those, P(A and B) is 0 and P(A or B) is just the sum.

      At least once in several tries

      The chance that something with probability p happens at least once in n independent tries is 1 − (1 − p)ⁿ. The chance of at least one six in four rolls of a die is 1 − (5/6)⁴ = 51.77%.

      Questions people ask

      If A has a 50% chance and B a 30% chance, what is the chance of both?

      15%, from 0.50 × 0.30. The chance of at least one is 65% and of neither is 35%.

      What is the probability of two heads in a row?

      25%. Each flip is 50%, and 0.5 × 0.5 = 0.25.

      How do I enter a probability given as a decimal?

      Multiply by 100: 0.25 is 25%.

      Related calculators

      Binomial Distribution CalculatorExact and cumulative binomial probabilities for n trials.Combination CalculatornCr — ways to choose r items from n when order does not matter.Permutation CalculatornPr — ordered arrangements of r items chosen from n.Expected Value CalculatorExpected value, variance and standard deviation of a set of outcomes.