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Probability and statistics

Addition rule (or)

The probability that A or B (or both) happens: add the two, then subtract the overlap so it isn't counted twice.

P(A∪B)=P(A)+P(B)−P(A∩B)
Mutually exclusive eventsP(A∪B)=P(A)+P(B)

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

See it move

ABrectangle: all outcomes
  • P(A ∪ B) = 0.5 + 0.4 − 0.2 = 0.7: the overlap is counted twice, so subtract it once.
  • Neither A nor B: 0.3.

What the symbols mean

P(A∪B)
the probability of A or B or both, P(A∪B)
P(A)
the probability of event A
P(B)
the probability of event B
P(A∩B)
the probability that both happen, P(A∩B)

When to use it

Use it for "or" questions: drawing a heart or a king, a student taking French or Spanish. If A and B cannot happen together (rolling a 2 or a 5 on one die), the overlap is 0 and you just add. Any probability you type can be a decimal or a percentage such as 30%.

Worked example

In a school, 50% of students study French, 40% study Spanish and 20% study both. What is the probability that a randomly chosen student studies French or Spanish?

  1. Start from the formula.

    P(A∪B)=P(A)+P(B)−P(A∩B)

  2. Substitute P(A) = 0.5, P(B) = 0.4, P(A∩B) = 0.2 and work it out.

    P(A∪B)=0.5+0.4−0.2
    =0.9−0.2
    =0.7

Answer P(French or Spanish) = 0.7

Why it works

Adding P(A) and P(B) counts the outcomes in both events twice, once in each. Subtracting P(A∩B) once leaves every outcome counted exactly once. The Venn diagram shows this.

Common mistakes

  • Adding without subtracting the overlap, which can give a "probability" above 1.
  • Assuming events are mutually exclusive when they are not: a card can be both a heart and a king.
  • Confusing P(A∪B) (or) with P(A∩B) (and).