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KS3 Maths · Year 7Chapter 12 · Probability
Year 7 · Chapter 12

Probability

By the end of this chapter you will be able to

  • Place events on the probability scale from 0 (impossible) to 1 (certain), and describe likelihood in words
  • Find relative frequency from an experiment, and use it to estimate a probability
  • Find the theoretical probability of an event from equally likely outcomes, and use the total probability of 1
  • List the sample space of one and two events, using a table or a list, and find probabilities from it
  • Use sets, unions, intersections and Venn diagrams to count outcomes
  • Add the probabilities of mutually exclusive events, and find the probability of an event not happening

You should already know

  • Fractions, decimals and percentages (Chapter 2)
  • Ratio and sharing (Chapter 8)
  • Venn and two-way tables from Year 6 (Year 6, Chapter 12)

Where this leads

  • Probability trees and independent events in Year 8
  • Conditional probability and sampling in Years 8 and 9

National Curriculum for England, Key Stage 3 mathematics, Year 7. Content: P1 record, describe and analyse outcome frequencies in experiments involving randomness, fairness, equal or unequal likelihood; the 0 to 1 scale, P2 the probabilities of all outcomes add up to 1, and the probability of an event not happening is 1 minus the probability of it happening, P3 enumerate sets, and unions and intersections, systematically with tables, grids and Venn diagrams, P4 sample spaces for single and combined events; equally likely and mutually exclusive outcomes; theoretical probability.

Exercise P  Do you remember?

Non-calculator
1
FHow many faces does a standard dice have, and how many numbers are on them?
2
FWrite 1/4 as a decimal.
3
FWhich is bigger, 0.3 or 1/3?
4
FWork out 1 − 0.4.
5
FWhich of 9, 11 and 15 is a prime number?
12.1

The probability scale and relative frequency

Year 7
P1
Video lesson: The probability scale and relative frequencyComing soon
A school fair has a spinner with coloured sections. Before the fair, the teacher spins the spinner 50 times and records each result. The counts show how often each colour came up, and they give an estimate of the chance of each colour.

A probability is a number from 0 (impossible) to 1 (certain). An event that is even (equally likely to happen or not) has a probability of 1/2. The 0 to 1 scale is marked with words: impossible, unlikely, even chance, likely and certain.

Relative frequency is the number of times an event happened, divided by the total number of trials. It is an estimate of the probability.

Key ideas

  • The probability scale runs from 0 (impossible) to 1 (certain).
  • A probability is never less than 0 or more than 1.
  • Relative frequency = number of times it happened ÷ number of trials.
  • Relative frequency gets closer to the probability as the number of trials gets bigger.
Worked example 12.1.1
A spinner lands on red 18 times in 50 spins. What is the relative frequency of red?

Method. Relative frequency = times it happened ÷ total trials.

  1. Write the fraction. 18 out of 50.
    18/50
  2. Simplify and write as a decimal. 18 ÷ 50 = 0.36.
    0.36
  3. Write the answer. Write the probability as a fraction in simplest form.
    0.36

Check. 0.36 × 50 = 18. ✓

Watch out. Divide the number of times by the total number of trials.

Worked example 12.1.2
A dice is thrown 60 times and a 6 comes up 12 times. Find the relative frequency of a 6.

Method. Relative frequency = times ÷ throws.

  1. Write the fraction. 12 out of 60.
    12/60
  2. Simplify. Divide the top and bottom by 12.
    1/5
  3. Write the answer. Write the probability as a fraction in simplest form.
    1/5

Check. 1/5 = 0.2, and 0.2 × 60 = 12. ✓

Watch out. Simplify the fraction.

Worked example 12.1.3
Describe where the probability of a fair dice landing on 7 goes on the scale.

Method. A dice has no 7 on it.

  1. Think about the outcomes. A standard dice has only 1 to 6, so 7 can never happen.
    no 7 on a dice
  2. Place it on the scale. An event that can never happen is impossible, at 0.
    0
  3. Write the answer. Write the probability as a fraction in simplest form.
    0, impossible

Check. The probability is 0. ✓

Watch out. Impossible is at the 0 end of the scale.

Watch out

Watch out: a probability cannot be more than 1. Saying "a 1.2 chance" is not possible, and 1 means certain.

Exam tip

Grown-up tip. Ask the pupil to flip a coin 20 times and record the relative frequency of heads, and to compare it with 0.5.

Exercise 12A  The probability scale and relative frequency

Non-calculator
1
FPut these words in order from least likely to most likely: certain, impossible, unlikely, evens, likely.
2
FWhat is the probability of rolling a 7 on a standard dice?
3
FWhat is the probability of rolling a number less than 7 on a standard dice?
4
FWhat is the probability of a fair coin landing on heads?
5
FA spinner lands on red 18 times in 50 spins. Find the relative frequency of red.
6
FWhere does a probability of 0.5 go on the scale?
7
FWhich is more likely: an event with probability 0.7, or one with probability 0.4?
8
PSA dice is rolled 30 times. An even number comes up 16 times. Find the relative frequency of an even number.
9
FGive a probability on the scale for each event: (a) an impossible event (b) a certain event (c) an event that is equally likely to happen or not
(a)(a) impossible
(b)(b) certain
(c)(c) equally likely to happen or not
10
PSAfter 200 spins, the relative frequency of blue is 0.25. How many times did blue come up?
11
PSIn 40 throws, a dice landed on 6 eight times. Estimate the probability of a 6 from these results.
12
RSam says a probability of 1.2 is possible for a weather forecast. Is he right? Explain.
12.2

Theoretical probability and the total of 1

Year 7
P2, P4
Video lesson: Theoretical probability and the total of 1Coming soon
A bag contains 3 red sweets and 5 blue sweets. If one sweet is taken without looking, each sweet is equally likely to be chosen. The chance of a red sweet can be worked out from the numbers, without any experiment.

The theoretical probability of an event is

P(event) = number of favourable outcomes ÷ total number of equally likely outcomes

The probabilities of all the outcomes add up to 1. So the probability that an event does not happen is 1 minus the probability that it does.

Key ideas

  • P(event) = favourable outcomes ÷ total equally likely outcomes.
  • The probabilities of all outcomes add up to 1.
  • P(not A) = 1 − P(A).
  • Count the outcomes, then write the probability as a fraction in simplest form.
Worked example 12.2.1
A bag has 3 red and 5 blue sweets. Find P(red).

Method. Favourable ÷ total.

  1. Count the total. 3 + 5 = 8 sweets.
    8
  2. Write the probability. Three red sweets out of 8.
    3/8
  3. Write the answer. Write the probability as a fraction in simplest form.
    3/8

Check. The blue probability is 5/8, and 3/8 + 5/8 = 1. ✓

Watch out. The total is every sweet.

Worked example 12.2.2
The probabilities of a spinner are 0.2 for A, 0.5 for B. Find the probability of C.

Method. The probabilities add to 1.

  1. Add the known probabilities. 0.2 + 0.5 = 0.7.
    0.7
  2. Subtract from 1. The rest must be C.
    1 − 0.7 = 0.3
  3. Write the answer. Write the probability as a fraction in simplest form.
    0.3

Check. 0.2 + 0.5 + 0.3 = 1. ✓

Watch out. The total is always 1.

Worked example 12.2.3
It is 35% likely to rain tomorrow. Find the probability that it does not rain.

Method. P(not rain) = 1 − P(rain).

  1. Write 35% as a decimal. 35% = 0.35.
    0.35
  2. Subtract from 1. 1 − 0.35.
    0.65
  3. Write the answer. Write the probability as a fraction in simplest form.
    0.65

Check. 0.65 + 0.35 = 1. ✓

Watch out. Use the rule for not happening.

Watch out

Watch out: the favourable outcomes go on the top, and the total number of outcomes on the bottom. Do not put the total on the top.

Exam tip

Grown-up tip. Ask the pupil to write the probabilities of all outcomes first, and check that they add to 1 before answering a question.

Exercise 12B  Theoretical probability

Non-calculator
1
FFind the probability of rolling a 6 on a fair dice.
2
FFind the probability of rolling an odd number on a fair dice.
3
FFind the probability of rolling a number greater than 4 on a fair dice.
4
FA bag has 4 red and 6 blue counters. Find P(red).
5
FA spinner has probabilities 0.2 for A, 0.5 for B and some probability for C. Find P(C).
6
FThe probability of rain is 0.35. Find the probability that it does not rain.
7
FA bag has 3 red and 7 blue marbles. Find the probability that a marble is not red.
8
PSA pack of 20 sweets has 5 cherry and 8 lemon. Find the probability of a sweet that is neither cherry nor lemon.
9
FA biased dice has probabilities 0.1, 0.2, 0.2, 0.2, 0.1 for the numbers 1 to 5. Find the probability of a 6.
10
PSA class of 35 pupils: the probability that a pupil walks to school is 0.6. How many pupils are expected to walk?
11
FFind the probability for each: (a) P(red) with 7 red and 7 blue counters (b) P(not 3) on a fair dice (c) P(vowel) for a letter chosen at random from the word MATHS
(a)(a) P(red), 7 red and 7 blue
(b)(b) P(not 3) on a fair dice
(c)(c) P(vowel) in MATHS
12
RExplain why P(heads) = 1/2 and P(tails) = 1/2 add up to 1.
12.3

Sample spaces for two events

Year 7
P4
Video lesson: Sample spaces for two eventsComing soon
Two friends each toss a coin. Their results can be listed as pairs: HH, HT, TH, TT. This list of all the possible outcomes is the sample space.

For two events, a sample space diagram (a table) shows all the outcomes. For two dice, there are 6 × 6 = 36 equally likely outcomes. Probabilities are found by counting the outcomes in the table that match the event.

123456
1234567
2345678
3456789
45678910
567891011
6789101112

Key ideas

  • List the sample space systematically, so that no outcome is missed or counted twice.
  • Two coins: 4 outcomes. Two dice: 36 outcomes.
  • Probability = matching outcomes ÷ total outcomes.
  • Check that the probabilities of all outcomes add to 1.
Worked example 12.3.1
Two coins are tossed. Find P(two heads).

Method. List the outcomes, then count.

  1. List the outcomes. HH, HT, TH, TT: four equally likely outcomes.
    4
  2. Count the matches. Only HH is two heads.
    1/4
  3. Write the answer. Write the probability as a fraction in simplest form.
    1/4

Check. 1 + 1 + 1 + 1 = 4 outcomes. ✓

Watch out. Write the sample space in order.

Worked example 12.3.2
Two dice are rolled and the scores are added. Find P(total = 7), using the table.

Method. Count the cells with 7.

  1. Find the total outcomes. Six by six: 36 cells.
    36
  2. Count the cells with 7. The diagonal from (1, 6) to (6, 1) has 6 cells with total 7.
    6
  3. Write the answer. Write the probability as a fraction in simplest form.
    6/36 = 1/6

Check. 6 ÷ 36 = 1/6. ✓

Watch out. Count the matching cells in the table.

Worked example 12.3.3
Two coins are tossed. Find P(at least one head).

Method. Count the outcomes with at least one head.

  1. List the outcomes. HH, HT and TH have at least one head.
    HH, HT, TH
  2. Probability. 3 out of 4 outcomes.
    3/4
  3. Write the answer. Write the probability as a fraction in simplest form.
    3/4

Check. Only TT has no head: 1 − 1/4 = 3/4. ✓

Watch out. Use the total outcomes as the bottom.

Watch out

Watch out: HT and TH are different outcomes when you toss two coins. Listing only "one head and one tail" as one outcome gives the wrong probability.

Exam tip

Grown-up tip. Ask the pupil to build the table on paper, shading the cells with a total of 7, and count them.

Exercise 12C  Sample spaces

Non-calculator
1
FTwo coins are tossed. How many outcomes are there?
2
FTwo coins are tossed. Find P(two heads).
3
FTwo coins are tossed. Find P(exactly one head).
4
FTwo dice are rolled. How many outcomes are there?
5
FTwo dice are rolled. Find P(total = 7).
6
FTwo dice are rolled. Find P(both numbers the same).
7
FTwo dice are rolled. Find P(total greater than 10).
8
PSA spinner has sections 1, 2, 3, 4 and a coin is tossed. Find P(3 and heads).
9
PSA spinner has 1, 2 and 3, and a dice is rolled. Find P(even on the dice and 1 on the spinner).
10
FThree coins are tossed. Find P(all tails).
11
PSA spinner has three equal sections, red, green and blue. It is spun twice. Find P(the same colour both times).
12
RExplain why P(total 2) is not the same as P(total 7) for two dice.
12.4

Sets, unions, intersections and Venn diagrams

Year 7
P3
Video lesson: Sets, unions, intersections and Venn diagramsComing soon
A club has 30 members. 18 play football, 12 play hockey, and 5 play both. A Venn diagram shows these groups and makes the counts clear.

A set is a collection of things. The intersection A ∩ B is the members in both sets. The union A ∪ B is the members in either set, counting each member once. The complement A′ is everything in the universal set that is not in A.

13575 neitherFootballHockey

Key ideas

  • A ∩ B (intersection): in both A and B.
  • A ∪ B (union): in A or B or both; count each member once.
  • A′ (complement): not in A.
  • In a Venn diagram, start from the middle (both sets), then fill in the rest.
Worked example 12.4.1
A = {1, 2, 3, 4} and B = {3, 4, 5, 6}. List A ∩ B.

Method. The numbers in both sets.

  1. Find the common numbers. Check each number of A to see if it is in B.
    3 and 4 are in both
  2. List. The intersection is {3, 4}.
    {3, 4}
  3. Write the answer. Write the probability as a fraction in simplest form.
    {3, 4}

Check. {3, 4} is in both lists. ✓

Watch out. List only the numbers in both sets.

Worked example 12.4.2
The club has 30 members: 18 football, 12 hockey, 5 both. How many play football or hockey (or both)?

Method. Union: add the two groups, then take away the overlap.

  1. Add the groups. 18 + 12 = 30.
    30
  2. Take away the overlap. The 5 who play both were counted twice.
    30 − 5 = 25
  3. Write the answer. Write the probability as a fraction in simplest form.
    25 members

Check. 25 + 5 = 30 with the overlap. ✓

Watch out. Count the overlap only once.

Worked example 12.4.3
U = {1, ..., 8}, A = {1, 3, 5, 7}. Find A′.

Method. A′ is everything in U that is not in A.

  1. Look at U. U has 1 to 8.
    1, 2, 3, 4, 5, 6, 7, 8
  2. Remove A. Take out 1, 3, 5 and 7.
    2, 4, 6, 8
  3. Write the answer. Write the probability as a fraction in simplest form.
    {2, 4, 6, 8}

Check. A and A′ together make all of U. ✓

Watch out. Use the universal set.

Watch out

Watch out: the intersection is the middle of the Venn diagram, and the union is everything. Do not add the overlap twice.

Exam tip

Grown-up tip. Ask the pupil to fill the middle of the Venn diagram first, then the parts that are only in one set, then the outside.

Exercise 12D  Sets and Venn diagrams

Non-calculator
1
FU = {1, ..., 10}, A = even numbers, B = multiples of 3. List A ∩ B.
2
FA = {1, 2, 3} and B = {3, 4}. List A ∪ B.
3
FIn a club of 30: 18 play football, 12 play hockey, 5 play both. How many play neither sport?
4
FIn a club of 30: 18 play football, 12 play hockey, 5 play both. Find P(plays both) for a member chosen at random.
5
PSIn the same club, find P(plays football or hockey).
6
FU = {1, ..., 8}, A = {1, 3, 5, 7}. List A′.
7
FA = {a, b, c, d} and B = {c, d, e}. List A ∩ B.
8
PSIn a class of 30, 14 like cats, 9 like dogs and 3 like both. How many like neither?
9
FIn a Venn diagram, A has 7 members in total and A ∩ B has 2. How many are in A only?
10
PSSet A has 8 elements, set B has 6, and the union has 12. How many are in both?
11
FSets: (a) if A has 9 members and B has 6 members, and 2 are in both, how many are in A or B? (b) how many are in A only in (a)?
(a)(a) A or B
(b)(b) A only
12
RExplain why P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
12.5

Combined events and mutually exclusive outcomes

Year 7
P4
Video lesson: Combined events and mutually exclusive outcomesComing soon
A spinner has sections 1 to 6. Landing on a 1 or a 2 are two outcomes that cannot happen together. When two outcomes cannot happen at the same time, they are mutually exclusive, and their probabilities can be added.

For mutually exclusive events A and B: P(A or B) = P(A) + P(B). For events that can happen together, subtract the overlap: P(A or B) = P(A) + P(B) − P(A and B).

Key ideas

  • Mutually exclusive: the events cannot happen at the same time.
  • P(A or B) = P(A) + P(B) when A and B are mutually exclusive.
  • P(not A) = 1 − P(A).
  • For a card: a king that is also a heart is counted in both, so subtract it once.
Worked example 12.5.1
A dice is rolled. Find P(1 or 2).

Method. The outcomes 1 and 2 are mutually exclusive.

  1. Add the probabilities. P(1) = 1/6 and P(2) = 1/6.
    1/6 + 1/6
  2. Add. Two sixths.
    2/6 = 1/3
  3. Write the answer. Write the probability as a fraction in simplest form.
    1/3

Check. 1/3 = 2 out of 6. ✓

Watch out. Only add when the events cannot both happen.

Worked example 12.5.2
A bag has red 0.3 and blue 0.5 probability. Find P(red or blue).

Method. Red and blue cannot both happen.

  1. Add. The two probabilities are mutually exclusive.
    0.3 + 0.5
  2. Add the decimals. Add them.
    0.8
  3. Write the answer. Write the probability as a fraction in simplest form.
    0.8

Check. The rest, 0.2, is another colour. ✓

Watch out. Add when the events cannot both happen.

Worked example 12.5.3
A card is drawn from a pack of 52. Find P(king or heart).

Method. Subtract the king of hearts, which is counted twice.

  1. Kings: 4, hearts: 13. Add the two groups.
    4 + 13 = 17
  2. Take away the overlap. The king of hearts is in both groups.
    17 − 1 = 16
  3. Write the answer. Write the probability as a fraction in simplest form.
    16/52 = 4/13

Check. 16 ÷ 52 = 4/13. ✓

Watch out. Count the overlap once.

Watch out

Watch out: only add probabilities when the events cannot both happen. The king and the heart can happen together (the king of hearts), so subtract the overlap.

Exam tip

Grown-up tip. Ask the pupil to say "can both happen?" before adding any probabilities.

Exercise 12E  Combined events

Non-calculator
1
FA dice is rolled. Find P(1 or 2).
2
FA dice is rolled. Find P(even or 5).
3
FA bag has red with probability 0.3 and blue with probability 0.5. Find P(red or blue).
4
FIn the bag above, find P(not red).
5
FA spinner has P(A) = 0.15, P(B) = 0.35 and P(C) = 0.4. Find P(D), if D is the only other outcome.
6
PSA card is drawn from a pack of 52. Find P(king or heart).
7
FA dice is rolled. Find P(prime or 6).
8
FA bag has 3 red, 5 blue and 2 green counters. Find P(not green).
9
FEvents A and B are mutually exclusive, with P(A) = 0.4 and P(B) = 0.35. Find P(A or B).
10
PSA spinner gives A, B or C with P(A) = 0.2 and P(B) = 0.45. Find P(A or B).
11
PSFind P(even or 3) on a fair dice.
12
RSam says that P(king or red card) = 4/52 + 26/52 = 30/52. Is Sam right? Explain.

Chapter summary

SkillMethod
The probability scale0 is impossible, 1 is certain, 1/2 is evens; probabilities are never less than 0 or more than 1
Relative frequencyTimes it happened ÷ total trials; an estimate of the probability
Theoretical probabilityFavourable outcomes ÷ total equally likely outcomes
Totals and not happeningAll probabilities add to 1; P(not A) = 1 − P(A)
Sample spacesTwo coins: 4 outcomes; two dice: 36 outcomes, shown in a table
Venn diagrams and setsIntersection: both; union: either, each counted once; complement: not in A
Mutually exclusive eventsP(A or B) = P(A) + P(B); otherwise subtract the overlap

Can you…?

  • I can place an event on the probability scale and describe its likelihood
  • I can find a relative frequency and use it to estimate a probability
  • I can find theoretical probabilities, and use the total of 1 and P(not A)
  • I can list a sample space for two events and find probabilities from it
  • I can use sets, unions, intersections and Venn diagrams to count outcomes
  • I can add the probabilities of mutually exclusive events

Key terms

probability
a number from 0 (impossible) to 1 (certain) that measures how likely an event is
relative frequency
the number of times an event happened divided by the number of trials
sample space
a list of all the possible outcomes of an experiment
mutually exclusive
two events that cannot both happen at the same time
intersection
the members that are in both sets, written A ∩ B
union
the members that are in either set or both, written A ∪ B
complement
everything in the universal set that is not in A, written A′
equally likely
outcomes that have the same chance of happening

Exercise 12Z  Check-up

Non-calculator
1
Find the probability of rolling a 3 on a fair dice.
2
A coin is tossed 50 times and lands on heads 22 times. Find the relative frequency of heads.
3
Two coins are tossed. Find P(exactly one head).
4
In a group, 12 like football, 9 like hockey and 4 like both. How many like football or hockey?
5
Find P(not 6) on a fair dice.
6
Find the probability of getting a total of 2 when two dice are rolled.
7
A bag has 3 white and 5 black counters. Find P(white).
8
Events are mutually exclusive: P(A) = 0.3 and P(B) = 0.45. Find P(C) if A, B and C are the only outcomes.
9
A card is drawn from a pack of 52. Find P(red card or king), with the overlap taken once.
10
The probability of success is 0.75. Out of 40 tries, how many successes are expected?

Total 21 marks. Allow about 21 minutes.

Exercise 12X  Practice test questions

Non-calculator
1
FFind the probability of rolling a 4 on a fair dice.[1 mark]
2
FThe probability of rain tomorrow is 0.35. Find the probability that it does not rain.[1 mark]
3
FTwo dice are rolled and the scores are added. Find P(total 10).[2 marks]
4
FIn a club of 25, 14 play maths, 10 play English and 4 play both. How many play neither?[2 marks]
5
FA spinner has P(A) = 0.5, P(B) = 0.1 and P(C) = 0.4. Find P(B or C).[2 marks]
6
FA bag contains 20 counters. The probability of picking a red counter is 0.4. How many red counters are in the bag?[2 marks]
7
FFind P(prime) for a fair dice.[2 marks]
8
FIn a group of 40 pupils, 25 like ice cream, 18 like cake and 10 like both. How many like neither?[3 marks]
9
FTwo spinners: the first has 1, 2 and 3, and the second has A and B. Find P(3 and B).[3 marks]
10
RExplain why P(A) + P(not A) = 1.[3 marks]
← Chapter 11: Perimeter, area, volume and PythagorasChapter 13: Statistics →

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