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Mutually Exclusive Events Calculator

Enter the probabilities of two exclusive events to get P(A or B).

How it is calculated

Enter the probability of each event as a decimal between 0 and 1 (not a percentage). Because the events cannot happen together, the chance that at least one occurs is the plain sum: P(A or B) = P(A) + P(B). The probability that neither occurs is the complement, 1 − P(A or B). Results are shown to four decimal places, and the combined probability is also given as a percentage. The tool returns nothing if either input is outside 0 to 1, or if the two probabilities add up to more than 1, because two truly mutually exclusive events cannot have a total above certainty. A sum of exactly 1 is accepted and leaves a chance of 0 for neither.

Worked example

Suppose P(A) = 0.3 and P(B) = 0.45. Add them: 0.3 + 0.45 = 0.75, so P(A or B) is 0.7500, or 75.00%. The remaining probability is 1 − 0.75 = 0.25, shown as 0.2500 for neither event. In words, there is a three-in-four chance that one of the two events happens, and a one-in-four chance that neither does.

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Common mistakes

  • Typing percentages such as 30 and 45. The inputs must be between 0 and 1, so 30 is rejected; enter 0.3 and 0.45.
  • Using the tool for events that can overlap, such as drawing a heart and drawing a king from one card. Those need P(A) + P(B) − P(A and B), which this tool does not apply.
  • Confusing mutually exclusive with independent. Exclusive events are strongly dependent: if A happens, B is certain not to.
  • Entering two probabilities that sum above 1 and expecting a capped answer. The tool gives no result, which signals that the events cannot really be mutually exclusive.
  • Reading the last figure as P(A and B). It is the probability that neither event occurs, while P(A and B) is always 0 here.

About this calculator

Two events are mutually exclusive (or disjoint) when they cannot both happen at the same time — like rolling a 3 and rolling a 5 on a single die roll. For such events, probability theory's addition rule simplifies to P(A or B) = P(A) + P(B), without the overlap-correcting subtraction term −P(A and B) that's needed for events that can co-occur, since that intersection is always zero.

This distinction is a foundational topic in introductory statistics and probability courses, and it matters in real applications like risk analysis (where two failure modes might be defined as mutually exclusive categories) and game design or gambling odds, where correctly identifying whether outcomes overlap determines whether you simply add probabilities or need the fuller inclusion-exclusion formula.

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Comments (1)

  • Editorial team

    The tool takes two events, but the same addition rule extends to any number of exclusive categories. Add them in steps: run the first two, then enter that combined probability as P(A) and the third event as P(B). For example, 0.2 and 0.3 give 0.5, and 0.5 with 0.1 gives 0.6. Because every step must stay at or below 1, a rejected entry is a quick hint that your categories overlap. How many outcomes are you trying to combine?

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