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Introduction to probability

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Cambridge O Level Mathematics (Syllabus D) 4024

Introduction to probability

The big idea: A probability is a number from 0 to 1 that measures how likely something is. With equally likely outcomes it is a fraction you can count: the outcomes you want over all the outcomes.

Summary — What This Chapter Covers

Probability is measured on a scale from 0 (impossible) to 1 (certain).
P(A) means the probability of event A; P(A′) means the probability of A not happening.
For equally likely outcomes, P(A) = number of outcomes in A ÷ total number of outcomes.
P(A′) = 1 − P(A).
Give a probability as a fraction, a decimal or a percentage. Information may come from tables, graphs or Venn diagrams.

1. The Probability Scale from 0 to 1

Syllabus 8.1 · Paper 1 & 2

Impossible to certain

Every probability lies between 0 and 1. A probability greater than 1, or less than 0, is always wrong.

Probability	Word	Example
0	impossible	rolling a 7 on an ordinary dice
between 0 and ½	unlikely	rolling a 6
½	even chance	a fair coin landing on heads
between ½ and 1	likely	rolling a number greater than 2
1	certain	rolling a number less than 7

Practice Question 1

A bag contains 3 red counters and 9 blue counters. One counter is taken at random. On a probability scale from 0 to 1, where should you mark P(red)? Describe this probability in words.

2. Probability Notation

Syllabus 8.1 · Paper 1 & 2

P(A) and P(A′)

P(A) is the probability that event A happens. P(A′) is the probability that A does not happen; A′ is read “A dash” or “not A”.

Write a probability as a fraction (3/10), a decimal (0.3) or a percentage (30%). Do not write it as a ratio (3 : 10) or in words such as “3 out of 10” or “3 in 10”: these lose the mark. Give fractions in their simplest form.

3. The Probability of a Single Event

Syllabus 8.1 · Paper 1 & 2

Count the outcomes

“At random” means every item has the same chance of being chosen, so all outcomes are equally likely. Then:

Equally likely outcomes P(A) = number of outcomes where A happens ÷ total number of possible outcomes

Worked Example

A bag contains 4 red, 5 blue and 3 green balls. One ball is taken at random. Find (a) P(blue) (b) P(not green) (c) P(yellow).

Practice Question 2

Each letter of the word STATISTICS is written on a separate card. One card is chosen at random. Find the probability that the letter is (a) T (b) a vowel.

4. The Probability of an Event Not Happening

Syllabus 8.1 · Paper 1 & 2

P(A′) = 1 − P(A)

An event either happens or it does not, so the two probabilities add up to 1. More generally, the probabilities of all the different possible outcomes of an experiment add up to 1. This is how you find a missing value in a probability table.

Complement rule P(A′) = 1 − P(A) e.g. P(B) = 0.8, so P(B′) = 1 − 0.8 = 0.2

Worked Example

A spinner can land on red, blue, green or yellow.

Colour	Red	Blue	Green	Yellow
Probability	0.3	0.26	x	3x

(a) Find x. (b) Find the probability that the spinner does not land on yellow.

5. Probabilities from Tables and Venn Diagrams

Syllabus 8.1 · Paper 1 & 2

Find the right count first

Memorise

Scale

Counting

Format

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Introduction to probability

Introduction to probability

Impossible to certain

P(A) and P(A′)

Count the outcomes

P(A′) = 1 − P(A)

Find the right count first

Scale

Counting

Format

Past papers that test Introduction to probability

More Mathematics D topics