Probability
Probability measures how likely something is, from 0 (impossible) to 1 (certain). Most questions use a few facts: probabilities of all the possible outcomes add up to 1, AND usually means multiply, and OR usually means add.
Part 1 of 3: Learn it
In short
- P(event) = number of ways it can happen ÷ total number of equally likely outcomes.
- P(not A) = 1 − P(A).
- Expected number of times = probability × number of trials.
Where this is in your specification
Spec points: DfE P1 to P9 (conditional probability, P9, is Higher tier)
| Board | Topic: Probability |
|---|---|
| DfE content | P1-P9 |
| AQA 8300 | Probability |
| Edexcel 1MA1 | Probability |
| OCR J560 | 13 Probability |
| Eduqas C300 | HP1 to HP9 |
| Cambridge IGCSE 0580 | E8.1 to E8.4 |
The basics
Write probabilities as fractions, decimals or percentages, never as "1 in 4" or "1 : 4". If a bag holds 5 red, 3 blue and 2 green counters, P(blue) = 3/10 and P(not green) = 1 − 2/10 = 8/10, which is 4/5.
If a table of probabilities has a missing value, use the fact that the whole table adds up to 1.
Sample spaces
List every outcome in a table or grid. Two fair dice give a 6 by 6 grid of 36 equally likely outcomes. A total of 9 comes from (3, 6), (4, 5), (5, 4) and (6, 3), so P(total 9) = 4/36 = 1/9.
Relative frequency and expected outcomes
- Relative frequency = number of times it happened ÷ number of trials. It estimates the probability from an experiment.
- More trials give a more reliable estimate.
- If a dice is rolled 120 times, you expect a 4 about 120 × 1/6 = 20 times.
Combined events and tree diagrams
For independent events, multiply along the branches to get P(A and B). Add the results of different routes for an "or" question. Each set of branches from one point adds up to 1. Without replacement (Higher), the numbers on the second set of branches change.
P(A) = 0.37. What is P(not A)?
Show the answer
1 − 0.37 = 0.63.
Part 2 of 3: See it worked
Worked examples
Example 1
A spinner lands on A, B, C or D with probabilities 0.35, 0.2, x and 2x. Find x, and how many times you expect it to land on C in 200 spins.
- 0.35 + 0.2 + x + 2x = 1
- 3x = 0.45, so x = 0.15
- Expected: 0.15 × 200 = 30
Answer: x = 0.15; about 30 times.
Example 2
The chance of rain on any day is 0.3, independent of other days. Find the probability it rains on exactly one of two days.
- Rain then dry: 0.3 × 0.7 = 0.21
- Dry then rain: 0.7 × 0.3 = 0.21
- Add the two routes: 0.21 + 0.21
Answer: 0.42.
Common mistakes
- Adding probabilities along tree branches instead of multiplying.
- Forgetting that the probabilities on each set of branches must add up to 1.
- Writing a probability greater than 1, or as a ratio.
- In a without-replacement question, keeping the same denominator on the second pick.
A dice shows a six 18 times in 80 rolls. Find the relative frequency of a six.
Show the answer
18 ÷ 80 = 0.225.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
P(A) = 0.37. What is P(not A)?
1 − 0.37 = 0.63.
A dice shows a six 18 times in 80 rolls. Find the relative frequency of a six.
18 ÷ 80 = 0.225.
Two fair coins are flipped. What is the probability of two heads?
½ × ½ = ¼.
Jobs that use this
- Actuary (s'ouvre dans un nouvel onglet)
- Insurance underwriter (s'ouvre dans un nouvel onglet)
- Data scientist (s'ouvre dans un nouvel onglet)
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Ces fiches sont en anglais car elles suivent les programmes d'examen britanniques.
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