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Probability and statistics

Multiplication rule for independent events (and)

When two events don't affect each other, the probability that both happen is the product of their probabilities.

P(A∩B)=P(A)·P(B)
General (any events)P(A∩B)=P(A)·P(B∣A)

Calculator

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

See it move

P(A) = 0.5P(B)
  • The overlap rectangle has area P(A) × P(B) = 0.5 × 0.25 = 0.125: that is P(A and B) when A and B are independent.

What the symbols mean

P(A∩B)
the probability that both A and B happen
P(A)
the probability of A
P(B)
the probability of B

When to use it

Use it for "and" questions about independent events: two coin tosses, a die and a spinner, separate machines failing. If the first event changes the odds of the second (cards drawn without replacement), multiply by the conditional probability instead, using the general form. The same rule extends to more events: the chance of 5 heads in a row is 0.55.

Worked example

A coin lands heads with probability 0.5 and a spinner lands on red with probability 0.25. What is the probability of heads and red?

  1. Start from the formula.

    P(A∩B)=P(A)⁢P(B)

  2. Substitute P(A) = 0.5, P(B) = 0.25 and work it out.

    P(A∩B)=0.5·0.25
    =0.125

Answer P(heads and red) = 0.125

Why it works

If A happens a fraction P(A) of the time and B happens a fraction P(B) of the time regardless of A, then B happens in a fraction P(B) of the cases where A happens. The unit-square picture shows the product as an area.

Common mistakes

  • Multiplying probabilities of dependent events as if they were independent.
  • Adding when the question says "and". "And" usually means multiply; "or" means add (and subtract the overlap).
  • Confusing independent with mutually exclusive: mutually exclusive events with non-zero probability can never be independent.