Probability: formula sheet

Key formulas

Classical probability

P(A) = m ÷ n

m = outcomes favourable to A, n = total outcomes. Valid only when all outcomes are equally likely.

Range of probability

0 ≤ P(A) ≤ 1

P = 0 for an impossible event and P = 1 for a certain event.

Complement rule

P(A') = 1 − P(A)

Use it for 'at least one' or 'not' questions.

Odds in favour and against

Odds in favour = m : (n − m); odds against = (n − m) : m

Convert odds a : b in favour to probability a ÷ (a + b).

Statistical probability

P(A) = limit of (f ÷ N) as N becomes very large

f = times A occurred, N = number of trials.

Mutually exclusive events

P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B)

Applies only when A and B cannot occur together.

Independent events

P(A ∩ B) = P(A) × P(B)

This is the test for independence.

Exhaustive and mutually exclusive set

P(E1) + P(E2) + … + P(Ek) = 1

Holds for events that are both exhaustive and mutually exclusive.

Addition theorem (general)

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Works for any two events A and B.

Mutually exclusive events

P(A ∪ B) = P(A) + P(B)

Use only when A and B cannot occur together, so P(A ∩ B) = 0.

Complementary event

P(A′) = 1 − P(A)

A and A′ are mutually exclusive, and P(A) + P(A′) = 1.

Neither A nor B

P(A′ ∩ B′) = 1 − P(A ∪ B)

This is De Morgan's law applied to probability.

At least one of A, B

P(at least one) = P(A ∪ B) = 1 − P(A′ ∩ B′)

Use the complement when it is easier to find P(neither).

Three events

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(B ∩ C) − P(A ∩ C) + P(A ∩ B ∩ C)

Add singles, subtract pairs, add back the triple overlap.

Conditional probability

P(A | B) = P(A ∩ B) ÷ P(B)

Valid only when P(B) > 0. Divide by the probability of the given event.

Multiplication theorem (general)

P(A ∩ B) = P(A) × P(B | A) = P(B) × P(A | B)

Works for any two events, dependent or independent.

Multiplication rule for independent events

P(A ∩ B) = P(A) × P(B)

Use only when the events are independent.

Test for independence

A and B are independent if P(A ∩ B) = P(A) × P(B), equivalently P(A | B) = P(A)

Check this when the question asks whether events are independent.

Three events (dependent)

P(A ∩ B ∩ C) = P(A) × P(B | A) × P(C | A ∩ B)

For independent events, multiply the three plain probabilities.

Complement of a conditional event

P(A' | B) = 1 − P(A | B)

The condition B stays the same on both sides.

Conditional probability

P(A | B) = P(A ∩ B) ÷ P(B)

Valid when P(B) > 0. This is the base of Bayes' theorem.

Multiplication rule

P(A ∩ B) = P(A) × P(B | A)

Gives the joint probability along a tree path.

Total probability theorem

P(B) = Σ P(Aᵢ) × P(B | Aᵢ)

A₁ to Aₙ must be mutually exclusive and exhaustive, so their priors add up to 1.

Bayes' theorem

P(Aᵢ | B) = P(Aᵢ) × P(B | Aᵢ) ÷ Σ P(Aⱼ) × P(B | Aⱼ)

Use when B has already happened and you need the probability of a cause Aᵢ. Needs P(B) > 0.

Check on posteriors

Σ P(Aᵢ | B) = 1

The posterior probabilities of all causes must add up to 1. Use this to check your work.

Expected value (discrete)

E(X) = Σ x·p(x) = x₁p₁ + x₂p₂ + ... + xₙpₙ

Multiply each value by its probability, then add.

Valid distribution

Σ p(x) = 1 and 0 ≤ p(x) ≤ 1

Use this to find a missing probability or an unknown constant.

Expectation of a function

E(X²) = Σ x²·p(x)

Square the values only, not the probabilities.

Expectation of a constant

E(c) = c

A fixed number has no chance element.

Multiplication by a constant and shift

E(aX + b) = a·E(X) + b

Holds for any constants a and b.

Addition property

E(X + Y) = E(X) + E(Y)

Always true for any two random variables, independent or not.

Multiplication property

E(XY) = E(X)·E(Y)

Valid only when X and Y are independent.

Variance using expectation

Var(X) = E(X²) − [E(X)]²

Useful when a question asks for variance after expectation.

Quick revision

• Probability always lies between 0 and 1, inclusive.
• P(A) + P(not A) = 1, so P(not A) = 1 − P(A).
• Mutually exclusive events: P(A ∪ B) = P(A) + P(B).
• General addition: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
• Independent events: P(A ∩ B) = P(A) × P(B).
• Conditional probability: P(A | B) = P(A ∩ B) ÷ P(B), for P(B) > 0.
• 'At least one' is often easiest as 1 − P(none).
• Bayes: P(A | B) = P(B | A) × P(A) ÷ P(B), where P(B) comes from total probability.
• Expectation: E(X) = Σ x × P(x).
• In a fair die there are 6 outcomes; in two dice 36; in a pack of cards 52.
• Mutually exclusive events with non-zero probabilities cannot be independent.

Common mistakes

• Treating mutually exclusive and independent as the same thing. Fix: Mutually exclusive means cannot occur together. Independent means one does not affect the other's probability. For non-zero probabilities, mutually exclusive events are dependent.
• Using m ÷ n when outcomes are not equally likely. Fix: Always list the underlying equally likely outcomes, such as the 36 pairs for two dice.
• Adding P(A) and P(B) without subtracting the overlap. Fix: Ask first: can both happen together? If yes, subtract P(A ∩ B).
• Treating mutually exclusive events as independent. Fix: Mutually exclusive means P(A ∩ B) = 0. Independent means P(A ∩ B) = P(A) × P(B). For events with non-zero probability, they cannot both be true.
• Using P(A) × P(B) when events are dependent, such as draws without replacement. Fix: Ask whether the first result changes the second. If yes, use P(A) × P(B | A) and update the counts.
• Dividing by the wrong probability in P(A | B), using P(A) instead of P(B). Fix: The event after the bar is the given one. Always divide by its probability.
• Using P(B | A) as the answer when the question asks for P(A | B). Fix: Underline the event that has already happened. That event goes after the bar in the required probability.
• Dividing by the prior probability or by the likelihood instead of by total P(B). Fix: The denominator is always the sum of the products of prior and likelihood over all causes.
• Dividing the sum of x·p by the number of values again. Fix: Since probabilities already add up to 1, E(X) = Σ x·p is the final answer. Do not divide again.
• Not checking that Σp = 1 before calculating. Fix: Add the probabilities first. Find any unknown from Σp = 1.

Exam tips

• Questions on types of events are usually conceptual. Learn the four definitions word for word in your own language, and practise telling mutually exclusive from independent.
• Many MCQs hide a complement. When you see 'at least one', calculate 1 − P(none) first.
• Know the pack of 52 cards cold, including 4 kings, 12 face cards and 26 red cards, so you do not waste time counting.
• There is no negative marking, so if time is short, eliminate options above 1 and guess among the rest.
• Look for the words "or", "either" and "at least one". They tell you to use the addition theorem.
• Check the wording "mutually exclusive" before choosing the formula. If it is absent, assume the overlap may exist.
• In "at least one" and "none" questions, compute the union first, then take 1 minus it where needed.
• For card and dice questions, count outcomes. It is faster than using the formula and avoids errors.