Probability

What you'll learn

How to describe chance using the probability scale from 0 to 1.
How to find probabilities by counting outcomes.
How to handle questions involving spinners, dice, bags, cards and tables.
How to find the probability of something not happening.

1. What is probability?

Probability is a way of measuring how likely something is to happen.

Definition

Probability

A probability is a number from 0 to 1 that shows how likely an event is to happen.

Probability 0 means the event is impossible.
Probability 1 means the event is certain.
Probability $12\frac{1}{2}$ means the event has an even chance.

Definition

Event and outcome

An outcome is one possible result, such as rolling a 4 on a dice.

An event is what you are interested in, such as rolling an even number.

Here is the probability scale:

Probability scale from 0 to 1 showing impossible, unlikely, even chance, likely and certain

Key Idea

The probability scale

Every probability must be between 0 and 1. If your answer is less than 0 or more than 1, something has gone wrong.

Worked example: placing a probability on a scale

A fair spinner has 4 equal sections. One section is labelled P, one is labelled Q, and two are labelled R. Mark the probability of landing on P on a probability scale.

Example

Spinner probability on a scale

Count the total number of equal sections.

There are 4 equal sections.
Count the number of sections labelled P.

There is 1 section labelled P.
Write the probability as “wanted outcomes over total outcomes”.
$P(P)=14P(\text{P})=\frac{1}{4}$
On the probability scale, place the cross halfway between 0 and $12\frac{1}{2}$ .

Common Mistake

Ignoring repeated labels

If a label appears more than once, count every section it appears in. Two sections labelled R means R has probability $24\frac{2}{4}$ , not $14\frac{1}{4}$ .

2. Finding probability by counting

When outcomes are equally likely, you can use the basic probability formula.

Definition

Equally likely

Outcomes are equally likely if each one has the same chance of happening. For example, on a fair dice, each number from 1 to 6 is equally likely.

The main formula is:

outcomes\text{Probability}=\frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}

A favourable outcome means an outcome that matches what the question asks for.

Worked example: choosing a number

A number is chosen at random from this list:

2, 4, 5, 7, 9, 10, 12, 15

Find the probability that the number chosen is 9.

Example

Choosing from a list

Count how many numbers are in the list.

There are 8 numbers.
Count how many of them are 9.

There is 1 number that is 9.
Write the probability as a fraction.
$18\frac{1}{8}$

Tip

At random

“At random” means you should treat each item as equally likely to be chosen, unless the question says otherwise.

3. Dice probabilities

An ordinary fair dice has 6 faces: 1, 2, 3, 4, 5 and 6.

Because it is fair, each face has probability $16\frac{1}{6}$ .

Worked example: even numbers on a dice

A fair dice is thrown once. Find the probability of getting an even number.

Example

Dice event: even number

List the possible dice outcomes.

The outcomes are 1, 2, 3, 4, 5 and 6.
Identify the even numbers.

The even numbers are 2, 4 and 6.
Count the favourable outcomes.

There are 3 favourable outcomes out of 6.
Write the probability and simplify if possible.
$36=12\frac{3}{6}=\frac{1}{2}$

Worked example: impossible dice outcome

A fair dice is thrown once. Find the probability of getting 8.

Example

Dice event: impossible outcome

Check whether 8 is on an ordinary dice.

An ordinary dice only has 1, 2, 3, 4, 5 and 6.
Since 8 cannot happen, there are no favourable outcomes.
$06=0\frac{0}{6}=0$
The probability is 0, so the event is impossible.

Common Mistake

Forgetting the dice only has 1 to 6

For an ordinary dice, outcomes like 0, 7, 8 or 10 are impossible. Their probability is 0.

4. Certain events and impossible events

Some probability questions are quicker if you recognise whether the event must happen or cannot happen.

Impossible means probability 0.
Certain means probability 1.

Worked example: certain dice event

A fair dice is thrown once. Find the probability that the number is less than 7.

Example

Certain dice event

List the possible dice outcomes.

The possible outcomes are 1, 2, 3, 4, 5 and 6.
Check the event “less than 7”.

Every possible dice outcome is less than 7.
So all 6 outcomes are favourable.
$66=1\frac{6}{6}=1$

Tip

Quick check

If the event includes every possible outcome, the probability is 1. If it includes no possible outcomes, the probability is 0.

5. Spinners, cards and repeated values

For fair spinners with equal sections, count the sections. For cards, count the actual cards — not just the different numbers.

Worked example: repeated numbers on cards

There are 7 cards with these numbers:

1, 6, 8, 6, 3, 8, 10

One card is chosen at random.

Find the probability that the card has the number 6 on it.

Example

Cards with repeated values

Count the total number of cards.

There are 7 cards.
Count how many cards show 6.

There are 2 cards showing 6.
Write the probability.
$27\frac{2}{7}$

Worked example: odd numbers on cards

Using the same cards, find the probability that the card has an odd number on it.

Example

Cards with an odd number

Identify the odd numbers in the list.

The odd numbers are 1 and 3.
Count how many odd-numbered cards there are.

There are 2 odd-numbered cards.
Put this over the total number of cards.
$27\frac{2}{7}$

Common Mistake

Counting types instead of items

If the cards are 1, 6, 8, 6, 3, 8, 10, there are 7 cards in total. Do not count only the different numbers.

6. Bags, boxes and tables

Many probability questions give counts of objects, such as pens, counters, sweets or chocolates.

The denominator is the total number of objects.

Worked example: finding a missing total category

A box contains 28 pens. There are 9 black pens and 6 green pens. The rest are red. One pen is chosen at random.

Find the probability that the pen is red.

Example

Pens in a box

Add the pens that are not red.
$9 + 6 = 15$
Subtract from the total to find the number of red pens.
$28 - 15 = 13$
Write the probability of choosing a red pen.
$1328\frac{13}{28}$

Worked example: using a table

A bag contains counters in these amounts:

Red: 6
Blue: 4
Yellow: 5
Green: 3

One counter is chosen at random. Find the probability that it is not blue.

Example

Not blue from a table

Find the total number of counters.
$6 + 4 + 5 + 3 = 18$
Count the counters that are not blue.

Red, yellow and green are not blue.
$6 + 5 + 3 = 14$
Write the probability.
$1418=79\frac{14}{18}=\frac{7}{9}$

Tip

Use the word not carefully

“Not blue” means every colour except blue. You can either add all the non-blue counters, or do total minus blue.

7. The probability of “not”

The probability of an event happening and the probability of it not happening always add to 1.

Key Idea

Complement rule

The probability of something not happening is:

happens1-\text{probability it happens}

For example, if the probability of winning is $38\frac{3}{8}$ , then the probability of not winning is $1−381-\frac{3}{8}$ .

Worked example: not winning with fractions

A player has probability $512\frac{5}{12}$ of winning a game. Find the probability that the player does not win.

Example

Probability of not winning

Start with the complement rule.
$1−5121-\frac{5}{12}$
Write 1 as $1212\frac{12}{12}$ .
$1212−512\frac{12}{12}-\frac{5}{12}$
Subtract the numerators.
$712\frac{7}{12}$

Worked example: not winning with decimals

The probability that a player wins a match is 0.65. Find the probability that the player does not win.

Example

Decimal complement

Use the rule that probabilities add to 1.
$1 - 0.65$
Subtract.
$0.35$

Common Mistake

Thinking not means impossible

“Not winning” does not mean probability 0. It means every outcome except winning, such as losing or drawing if a draw is possible.

8. Completing probability tables

Sometimes you are given probabilities for several outcomes and asked to find the missing one.

All probabilities in a complete set add to 1.

Worked example: missing probability

A counter is picked from a bag. The probabilities are:

Red: 0.25
Blue: 0.35
Yellow: 0.15
Green: unknown

Find the probability of green.

Example

Completing probabilities

Add the probabilities you know.
$0.25 + 0.35 + 0.15 = 0.75$
Subtract from 1 to find the missing probability.
$1 - 0.75 = 0.25$
So the probability of green is 0.25.

Common Mistake

Only add probabilities for all possible outcomes

You can only subtract from 1 when the listed outcomes cover every possible result. For colours in a bag, this works if every counter is one of those colours.

Exam technique

In the exam

Count the total number of equally likely outcomes first — this is usually the denominator.
Count the outcomes the question asks for — this is usually the numerator.
For “not” questions, either count everything except that outcome or subtract from 1.
Check your answer is between 0 and 1, and simplify fractions when it is easy.

Self review

Check yourself

Can you explain why the probability of rolling a number greater than 6 on a fair dice is 0?
If a bag has 5 red counters and 15 counters altogether, what fraction of the counters are red?
If the probability of rain is 0.4, what calculation gives the probability of no rain?

Probability

What you'll learn

How to describe chance using the probability scale from 0 to 1.
How to find probabilities by counting outcomes.
How to handle questions involving spinners, dice, bags, cards and tables.
How to find the probability of something not happening.

1. What is probability?

Probability is a way of measuring how likely something is to happen.

Definition

Probability

A probability is a number from 0 to 1 that shows how likely an event is to happen.

Probability 0 means the event is impossible.
Probability 1 means the event is certain.
Probability $12\frac{1}{2}$ means the event has an even chance.

Definition

Event and outcome

An outcome is one possible result, such as rolling a 4 on a dice.

An event is what you are interested in, such as rolling an even number.

Here is the probability scale:

Probability scale from 0 to 1 showing impossible, unlikely, even chance, likely and certain

Key Idea

The probability scale

Every probability must be between 0 and 1. If your answer is less than 0 or more than 1, something has gone wrong.

Worked example: placing a probability on a scale

A fair spinner has 4 equal sections. One section is labelled P, one is labelled Q, and two are labelled R. Mark the probability of landing on P on a probability scale.

Example

Spinner probability on a scale

Count the total number of equal sections.

There are 4 equal sections.
Count the number of sections labelled P.

There is 1 section labelled P.
Write the probability as “wanted outcomes over total outcomes”.
$P(P)=14P(\text{P})=\frac{1}{4}$
On the probability scale, place the cross halfway between 0 and $12\frac{1}{2}$ .

Common Mistake

Ignoring repeated labels

If a label appears more than once, count every section it appears in. Two sections labelled R means R has probability $24\frac{2}{4}$ , not $14\frac{1}{4}$ .

2. Finding probability by counting

When outcomes are equally likely, you can use the basic probability formula.

Definition

Equally likely

Outcomes are equally likely if each one has the same chance of happening. For example, on a fair dice, each number from 1 to 6 is equally likely.

The main formula is:

outcomes\text{Probability}=\frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}

A favourable outcome means an outcome that matches what the question asks for.

Worked example: choosing a number

A number is chosen at random from this list:

2, 4, 5, 7, 9, 10, 12, 15

Find the probability that the number chosen is 9.

Example

Choosing from a list

Count how many numbers are in the list.

There are 8 numbers.
Count how many of them are 9.

There is 1 number that is 9.
Write the probability as a fraction.
$18\frac{1}{8}$

Tip

At random

“At random” means you should treat each item as equally likely to be chosen, unless the question says otherwise.

3. Dice probabilities

An ordinary fair dice has 6 faces: 1, 2, 3, 4, 5 and 6.

Because it is fair, each face has probability $16\frac{1}{6}$ .

Worked example: even numbers on a dice

A fair dice is thrown once. Find the probability of getting an even number.

Example

Dice event: even number

List the possible dice outcomes.

The outcomes are 1, 2, 3, 4, 5 and 6.
Identify the even numbers.

The even numbers are 2, 4 and 6.
Count the favourable outcomes.

There are 3 favourable outcomes out of 6.
Write the probability and simplify if possible.
$36=12\frac{3}{6}=\frac{1}{2}$

Worked example: impossible dice outcome

A fair dice is thrown once. Find the probability of getting 8.

Example

Dice event: impossible outcome

Check whether 8 is on an ordinary dice.

An ordinary dice only has 1, 2, 3, 4, 5 and 6.
Since 8 cannot happen, there are no favourable outcomes.
$06=0\frac{0}{6}=0$
The probability is 0, so the event is impossible.

Common Mistake

Forgetting the dice only has 1 to 6

For an ordinary dice, outcomes like 0, 7, 8 or 10 are impossible. Their probability is 0.

4. Certain events and impossible events

Some probability questions are quicker if you recognise whether the event must happen or cannot happen.

Impossible means probability 0.
Certain means probability 1.

Worked example: certain dice event

A fair dice is thrown once. Find the probability that the number is less than 7.

Example

Certain dice event

List the possible dice outcomes.

The possible outcomes are 1, 2, 3, 4, 5 and 6.
Check the event “less than 7”.

Every possible dice outcome is less than 7.
So all 6 outcomes are favourable.
$66=1\frac{6}{6}=1$

Tip

Quick check

If the event includes every possible outcome, the probability is 1. If it includes no possible outcomes, the probability is 0.

5. Spinners, cards and repeated values

For fair spinners with equal sections, count the sections. For cards, count the actual cards — not just the different numbers.

Worked example: repeated numbers on cards

There are 7 cards with these numbers:

1, 6, 8, 6, 3, 8, 10

One card is chosen at random.

Find the probability that the card has the number 6 on it.

Example

Cards with repeated values

Count the total number of cards.

There are 7 cards.
Count how many cards show 6.

There are 2 cards showing 6.
Write the probability.
$27\frac{2}{7}$

Worked example: odd numbers on cards

Using the same cards, find the probability that the card has an odd number on it.

Example

Cards with an odd number

Identify the odd numbers in the list.

The odd numbers are 1 and 3.
Count how many odd-numbered cards there are.

There are 2 odd-numbered cards.
Put this over the total number of cards.
$27\frac{2}{7}$

Common Mistake

Counting types instead of items

If the cards are 1, 6, 8, 6, 3, 8, 10, there are 7 cards in total. Do not count only the different numbers.

6. Bags, boxes and tables

Many probability questions give counts of objects, such as pens, counters, sweets or chocolates.

The denominator is the total number of objects.

Worked example: finding a missing total category

A box contains 28 pens. There are 9 black pens and 6 green pens. The rest are red. One pen is chosen at random.

Find the probability that the pen is red.

Example

Pens in a box

Add the pens that are not red.
$9 + 6 = 15$
Subtract from the total to find the number of red pens.
$28 - 15 = 13$
Write the probability of choosing a red pen.
$1328\frac{13}{28}$

Worked example: using a table

A bag contains counters in these amounts:

Red: 6
Blue: 4
Yellow: 5
Green: 3

One counter is chosen at random. Find the probability that it is not blue.

Example

Not blue from a table

Find the total number of counters.
$6 + 4 + 5 + 3 = 18$
Count the counters that are not blue.

Red, yellow and green are not blue.
$6 + 5 + 3 = 14$
Write the probability.
$1418=79\frac{14}{18}=\frac{7}{9}$

Tip

Use the word not carefully

“Not blue” means every colour except blue. You can either add all the non-blue counters, or do total minus blue.

7. The probability of “not”

The probability of an event happening and the probability of it not happening always add to 1.

Key Idea

Complement rule

The probability of something not happening is:

happens1-\text{probability it happens}

For example, if the probability of winning is $38\frac{3}{8}$ , then the probability of not winning is $1−381-\frac{3}{8}$ .

Worked example: not winning with fractions

A player has probability $512\frac{5}{12}$ of winning a game. Find the probability that the player does not win.

Example

Probability of not winning

Start with the complement rule.
$1−5121-\frac{5}{12}$
Write 1 as $1212\frac{12}{12}$ .
$1212−512\frac{12}{12}-\frac{5}{12}$
Subtract the numerators.
$712\frac{7}{12}$

Worked example: not winning with decimals

The probability that a player wins a match is 0.65. Find the probability that the player does not win.

Example

Decimal complement

Use the rule that probabilities add to 1.
$1 - 0.65$
Subtract.
$0.35$

Common Mistake

Thinking not means impossible

“Not winning” does not mean probability 0. It means every outcome except winning, such as losing or drawing if a draw is possible.

8. Completing probability tables

Sometimes you are given probabilities for several outcomes and asked to find the missing one.

All probabilities in a complete set add to 1.

Worked example: missing probability

A counter is picked from a bag. The probabilities are:

Red: 0.25
Blue: 0.35
Yellow: 0.15
Green: unknown

Find the probability of green.

Example

Completing probabilities

Add the probabilities you know.
$0.25 + 0.35 + 0.15 = 0.75$
Subtract from 1 to find the missing probability.
$1 - 0.75 = 0.25$
So the probability of green is 0.25.

Common Mistake

Only add probabilities for all possible outcomes

You can only subtract from 1 when the listed outcomes cover every possible result. For colours in a bag, this works if every counter is one of those colours.

Exam technique

In the exam

Count the total number of equally likely outcomes first — this is usually the denominator.
Count the outcomes the question asks for — this is usually the numerator.
For “not” questions, either count everything except that outcome or subtract from 1.
Check your answer is between 0 and 1, and simplify fractions when it is easy.

Self review

Check yourself

Can you explain why the probability of rolling a number greater than 6 on a fair dice is 0?
If a bag has 5 red counters and 15 counters altogether, what fraction of the counters are red?
If the probability of rain is 0.4, what calculation gives the probability of no rain?

Probability

What you'll learn

1. What is probability?

Worked example: placing a probability on a scale

2. Finding probability by counting

Worked example: choosing a number

3. Dice probabilities

Worked example: even numbers on a dice

Worked example: impossible dice outcome

4. Certain events and impossible events

Worked example: certain dice event

5. Spinners, cards and repeated values

Worked example: repeated numbers on cards

Worked example: odd numbers on cards

6. Bags, boxes and tables

Worked example: finding a missing total category

Worked example: using a table

7. The probability of “not”

Worked example: not winning with fractions

Worked example: not winning with decimals

8. Completing probability tables

Worked example: missing probability

Recap questions

You've reached the end

Probability Revision Guide

Probability

What you'll learn

1. What is probability?

Worked example: placing a probability on a scale

2. Finding probability by counting

Worked example: choosing a number

3. Dice probabilities

Worked example: even numbers on a dice

Worked example: impossible dice outcome

4. Certain events and impossible events

Worked example: certain dice event

5. Spinners, cards and repeated values

Worked example: repeated numbers on cards

Worked example: odd numbers on cards

6. Bags, boxes and tables

Worked example: finding a missing total category

Worked example: using a table

7. The probability of “not”

Worked example: not winning with fractions

Worked example: not winning with decimals

8. Completing probability tables

Worked example: missing probability

Recap questions

You've reached the end

Probability Revision Guide