S5.1StatisticsFoundation

Calculating Probabilities

Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). For equally likely outcomes it is just favourable outcomes over total outcomes — the foundation for everything that follows.

20 min Probability

Edexcel AS Level Maths: 5.1 Calculating ProbabilitiesCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Use the equally-likely-outcomes formula
Work with the sample space
Use the complement rule
Apply probability to simple experiments

The basic formula

When outcomes are equally likely, the probability of an event is the number of favourable outcomes divided by the total number of outcomes. The full set of possible outcomes is the $\textbf{sample space}$ .

P(A) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}

For equally likely outcomes.

Example 1A fair die is rolled. Find P(even number).

1Favourable: 2, 4, 6 (3 outcomes). Total: 6.

Answer

\tfrac{1}{2}

The probability scale

All probabilities lie between $0$ and $1$ . An impossible event has probability $0$ ; a certain event has probability $1$ . The probabilities of all outcomes in a sample space add to $1$ .

0 \le P(A) \le 1

Probabilities never go outside this range.

The complement

The $\textbf{complement}$ $A'$ is “ $A$ does not happen”. Since something either happens or it does not, the two probabilities add to $1$ .

P(A') = 1 - P(A)

Often the quickest route to an answer.

Tip — If “at least one” is awkward, find P(none) and subtract from 1.

Formula recap

P(A) = \dfrac{\text{favourable}}{\text{total}}

Equally likely outcomes.

0 \le P(A) \le 1

Probability scale.

P(A') = 1 - P(A)

Complement rule.

Common mistakes to avoid

Giving a probability greater than 1.

Every probability is between 0 and 1.

Computing “at least one” the long way.

Use 1 − P(none) via the complement.

Key takeaways

P(A) = favourable ÷ total for equally likely outcomes.
All probabilities lie in [0, 1] and sum to 1 over the sample space.
Complement: P(A') = 1 − P(A).

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Calculating Probabilities. Use the feedback to decide what needs another look.

2 past-paper questions · 2 marks

Practise this topic

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.