Probability

Maths · Permutation, combination and probability. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

Test this topic free All Maths topics

The idea

The sample space S is the set of all possible outcomes of an experiment, and an event E is a subset of S. When all outcomes are equally likely, the probability of E is the number of outcomes in E divided by the number of outcomes in S. It always lies from 0 (impossible) to 1 (certain).

For "A or B" you add, but outcomes that are in both events must not be counted twice. Events that cannot happen together are called mutually exclusive, and for them there is nothing to subtract.

For "A and B" with independent events, where one does not affect the other, you multiply the probabilities.

Rules to remember

P(E) = n(E)/n(S) = (number of favourable outcomes) ÷ (total number of equally likely outcomes)
0 ≤ P(E) ≤ 1, and P(not E) = 1 − P(E)
Mutually exclusive events: P(A ∪ B) = P(A) + P(B)
Events not mutually exclusive: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Independent events: P(A ∩ B) = P(A) × P(B)
n(S): one die 6, two dice 36, two coins 4, a pack of cards 52

Common mistake

Students add P(A) and P(B) for "or" without checking for overlap. If some outcomes are in both events, subtract P(A ∩ B) once, or the answer is too large.

Worked example

One number is chosen at random from the whole numbers 1 to 20. Find the probability that it is a multiple of 3 or a multiple of 5.

Multiples of 3: 3, 6, 9, 12, 15, 18, so P(A) = 6/20. Multiples of 5: 5, 10, 15, 20, so P(B) = 4/20.
15 is in both events, so P(A ∩ B) = 1/20.
P(A ∪ B) = 6/20 + 4/20 − 1/20 = 9/20.

Answer: 9/20

Ask about this

Type what you are stuck on. The helper finds the explanation, worked examples and the notes to read, free and instant.

Keep going

← Combinations Mathematical induction →Take a Maths test