Probability Updated November 2026

Probability rules

Three rules cover most problems. Addition: P(A or B) = P(A) + P(B) - P(A and B). Multiplication for independent events: P(A and B) = P(A) * P(B). Complement: P(not A) = 1 - P(A), which is the fastest way to get the probability of at least one.

Syntax

P(A or B) = P(A) + P(B) - P(A and B)

Part	What it is
`P(A and B)`	the probability both happen; 0 when A and B are mutually exclusive
`P(not A)`	1 - P(A), the complement

Examples

One die: P(even or greater than 4)
P(even) = 3/6, P(> 4) = 2/6, P(even and > 4) = 1/6 (the 6)
3/6 + 2/6 - 1/6

Result: 4/6 = 2/3

Two coins, independent: P(both heads)
1/2 * 1/2

Result: 1/4

Two dice: P(at least one six)
1 - P(no six on either) = 1 - (5/6)^2 = 1 - 25/36

Result: 11/36 = 0.3056

Mutually exclusive: P(roll a 1 or a 2)
1/6 + 1/6

Result: 2/6 = 1/3

Common mistakes

Adding P(A) and P(B) without subtracting the overlap; that double counts outcomes that belong to both.
Multiplying probabilities of events that are not independent; use P(A) * P(B | A) instead.
Computing at least one by adding single-event probabilities (1/6 + 1/6 for two dice); use the complement 1 - (5/6)^2.

Practice Probability rules

These drills check your answer on live data, free, no account needed.

Addition rule: A or BIn a survey of adults, 22% buy a coffee, 55% buy a pastry, and 15% do both. If one respondent is picked at random, what is the probability they buy a coffee or buy a pastry (or both)? Round to 4 decimal places.
Multiplication rule: all succeedConsider a data pipeline with three stages that fail independently. Each part works with probability 0.9, 0.85, 0.99 respectively, independently of the others. Everything must work for the whole thing to succeed. What is the probability of success? Round to 4 decimal places.
At least one, by the complementTake 5 loan files in an audit. Each one has a documentation error with probability 0.01, independently of the others. What is the probability that at least one of the 5 has a documentation error? Round to 4 decimal places.

Questions

What is the difference between mutually exclusive and independent?

Mutually exclusive events cannot both happen, so P(A and B) = 0. Independent events do not affect each other, so P(A and B) = P(A) * P(B). Two events with nonzero probability cannot be both.

How do I calculate the probability of at least one success?

Take 1 minus the probability of zero successes. For n independent tries with success probability p: 1 - (1 - p)^n.

Can a probability be greater than 1?

No. If an addition-rule calculation goes above 1 you forgot to subtract the overlap.

Conditional probabilityHow to calculate conditional probability P(A|B) = P(A and B) / P(B), read it from a two-way table, check independence and avoid confusing P(A|B) with P(B|A).
Bayes' theoremBayes theorem updates a probability with new evidence. The formula, a medical screening example with exact numbers, the natural frequency shortcut and errors.
Binomial distributionThe binomial formula P(X = k) = nCk p^k (1 - p)^(n - k), its mean np and standard deviation sqrt(np(1 - p)), exact worked examples and Excel BINOM.DIST.
Permutations and combinationsWhen order matters use permutations nPr = n! / (n - r)!; otherwise combinations nCr = n! / (r! (n - r)!). Examples plus Excel PERMUT and COMBIN.