Basic Probability Notes

Cambridge IGCSE Maths: Revision notes

Key facts

Probability runs from 0 (impossible) to 1 (certain). It can be a fraction, decimal or percentage.
For equally likely outcomes, $P(\text{event})=\dfrac{\text{favourable outcomes}}{\text{total outcomes}}$ .
$P(\text{not event})=1-P(\text{event})$ , written $P(A')=1-P(A)$ .
Fair means every outcome is equally likely; biased means they are not.
The probabilities of all outcomes add up to 1.

The probability scale

Probability measures likelihood from 0 (impossible) to 1 (certain).

Probability measures how likely an event is. $P=0$ means it cannot happen, $P=1$ means it is certain and $P=0.5$ means it is equally likely to happen or not. Write probabilities as fractions, decimals or percentages. A probability can never be less than 0 or more than 1.

Which of these cannot be a probability?

Single events

Divide the number of favourable outcomes by the total number of outcomes.

For equally likely outcomes, count the outcomes you want and divide by all possible outcomes. For a fair die, $P(4)=\dfrac16$ . Give the answer in its simplest form unless told otherwise.

5 red and 3 blue balls: 3 favourable outcomes out of 8, so P(blue) = 3/8

Probability $\dfrac{\text{favourable outcomes}}{\text{total outcomes}}$

Worked example

A bag has 5 red and 3 blue balls. Find $P(\text{blue})$ .

1
Favourable (blue): 3.
2
Total: $5+3=8$ .

A bag has 5 red and 3 blue balls. What is $P(\text{red})$ ?

Complementary events

The probability of an event not happening is 1 minus the probability that it does.

The complement of an event is "the event not happening". All outcomes add up to 1, so $P(\text{not event})=1-P(\text{event})$ . At Extended this is written $P(A')=1-P(A)$ .

The two probabilities add up to 1: P(rain) = 0.3, so P(no rain) = 1 − 0.3 = 0.7

$P(A')$ $1-P(A)$

Worked example

$P(\text{rain tomorrow})=0.3$ . Find $P(\text{no rain})$ .

1
Use $P(\text{not rain})=1-P(\text{rain})$ .
2
$1-0.3$ .

$P(A)=0.35$ . What is $P(A')$ ?

Fair, biased and random

Exam wording uses these precisely: fair means equal chances, biased does not.

A fair coin, die or spinner gives every outcome an equal chance. A biased one does not, so you cannot assume $\frac16$ for each number on a biased die. A random event is one whose outcome cannot be predicted with certainty in advance.

Fair die
Biased die

A fair die gives every number probability 1/6 (0.167); the biased die is illustrative values and each total is 1

Fair

Every outcome equally likely
Coin, die, spinner

Biased

Outcomes not equally likely
Use the given probabilities

Random

Outcome cannot be predicted in advance

A biased die has $P(6)=0.3$ . What is the probability of not rolling a 6?

Try an exam question

A bag contains 6 red counters and 4 green counters. A counter is taken at random. (a) Write down the probability that the counter is green. Give your answer as a fraction in its simplest form. (b) Write down the probability that the counter is not green.

[3 marks]

That's the notes covered.

Carry on to the next subtopic.