KS3 Mathematics
D4 Probability
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Contents
D4 Probability
D4.1 The language of probability
D4.2 The probability scale
D4.3 Calculating probability
D4.4 Probability diagrams
D4.5 Experimental probability
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The language of probability
Probability is a measurement of the chance or likelihood of
an event happening.
Words that we might use to describe probabilities include:
unlikely
possible
50-50
chance
certain
likely
poor
chance
very
likely
impossible
probable
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even
chance
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Fair games
A game is played with marbles in a bag.
One of the following bags is chosen for the game. The
teacher then pulls a marble at random from the chosen bag:
bag a
bag b
bag c
If a red marble is pulled out of the bag, the girls get a point.
If a blue marble is pulled out of the bag, the boys get a point.
Which would be the fair bag to use?
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Fair games
A game is fair if all the players have an
equal chance of winning.
Which of the following games are fair?
A dice is thrown. If it lands on a prime number team A get
a point, if it doesn’t team B get a point.
There are three prime numbers (2, 3 and 5) and three
non-prime numbers (1, 4 and 6).
Yes, this game is fair.
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Fair games
Nine cards numbered 1 to 9 are used and a card is drawn
at random.
If a multiple of 3 is drawn team A get a point.
If a square number is drawn team B get a point.
If any other number is drawn team C get a point.
There are three multiples of 3 (3, 6 and 9).
There are three square numbers (1, 4 and 9).
There are four numbers that are neither square nor
multiples of 3 (2, 5, 7 and 8).
No, this game is not fair. Team C is more likely to win.
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Fair games
A spinner has five equal sectors numbered 1 to 5.
The spinner is spun many times.
5
1
If the spinner stops on an even
number team A gets 3 points.
4
2
If the spinner stops on an odd
3
number team B gets 2 points.
Suppose the spinner is spun 50 times.
We would expect the spinner to stop on an even number 20
times and on an odd number 30 times.
Team A would score 20 × 3 points = 60 points
Team B would score 30 × 2 points = 60 points
Yes, this game is fair.
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Scratch cards
Scratch off a £ sign
and win £10!
£
no
win
£
no
win
no
win
no
win
no
win
£
no
win
no
win
£
£
no no
win win
no no
win win
£
no no no no
win win win win
no no
win win
£
no
win
no no no no
win win win win
no
win
£ £
no
win
£
no no
win win
£
You are only allowed to scratch off one square and you can’t
see what is behind any of the squares.
Which of the scratch cards is most likely to win a prize?
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Bags of counters
Choose a blue counter
and win a prize!
bag a
bag b
bag c
You are only allowed to choose one counter at random from
one of the bags.
Which of the bags is most likely to win a prize?
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Probability statements
Statements involving probability are often incorrect or
misleading. Discuss the following statements:
The number 18 is has been
drawn the most often in the
national lottery so I’m more
likely to win if I choose it.
I’m so unlucky. If
I roll this dice I’ll
never get a six.
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I’ve just thrown four heads in a
row so I’m much less likely to
get a head on my next throw.
There are two choices for lunch,
pizza or curry. That means that
there is a 50% chance that the
next person will choose pizza.
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Contents
D4 Probability
D4.1 The language of probability
D4.2 The probability scale
D4.3 Calculating probability
D4.4 Probability diagrams
D4.5 Experimental probability
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The probability scale
The chance of an event happening can be shown on a
probability scale.
Meeting
with King
Henry VIII
A day of the
week starting
with a T
The next baby
born being a
boy
Getting
homework
this lesson
A square
having four
right angles
impossible
unlikely
even chance
likely
certain
Less likely
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More likely
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The probability scale
We measure probability on a scale from 0 to 1.
If an event is impossible or has no probability of occurring
then it has a probability of 0.
If an event is certain it has a probability of 1.
This can be shown on the probability scale as:
0
impossible
½
even chance
1
certain
Probabilities are written as fractions, decimal and, less often,
as percentages between 0 and 1.
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The probability scale
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Contents
D4 Probability
D4.1 The language of probability
D4.2 The probability scale
D4.3 Calculating probability
D4.4 Probability diagrams
D4.5 Experimental probability
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Higher or lower
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Listing possible outcomes
When you roll a fair dice you are equally likely to get one
of six possible outcomes:
1
6
1
6
1
6
1
6
1
6
1
6
Since each number on the dice is equally likely the
probability of getting any one of the numbers is 1 divided
1
by 6 or
.
6
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Calculating probability
What is the probability of the following events?
1) A coin landing tails up?
P(tails) =
1
2
2) This spinner stopping on
the red section?
1
P(red) =
4
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3) Drawing a seven of hearts
from a pack of 52 cards?
P(7 of
)=
1
52
4) A baby being born on a
Friday?
P(Friday) =
1
7
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Calculating probability
If the outcomes of an event are equally likely then we can
calculate the probability using the formula:
Probability of an event =
Number of successful outcomes
Total number of possible outcomes
For example, a bag contains 1 yellow,
3 green, 4 blue and 2 red marbles.
What is the probability of pulling a green
marble from the bag without looking?
3
P(green) =
or 0.3 or 30%
10
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Calculating probability
This spinner has 8 equal divisions:
What is the probability of the
spinner landing on
a) a red sector?
b) a blue sector?
c) a green sector?
2
1
=
8
4
1
b) P(blue) =
8
4
1
c) P(green) =
=
8
2
a) P(red) =
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Calculating probability
A fair dice is thrown. What is the probability of getting
a) a 2?
b) a multiple of 3?
c) an odd number?
d) a prime number?
e) a number bigger than 6?
f) an integer?
a) P(2) = 1
6
2
1
b) P(a multiple of 3) =
=
6
3
3
1
c) P(an odd number) =
=
6
2
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Calculating probability
A fair dice is thrown. What is the probability of getting
a) a 2?
b) a multiple of 3?
c) an odd number?
d) a prime number?
e) a number bigger than 6?
f) an integer?
d) P(a prime number) =
3
1
=
6
2
Don’t write
0
6
e) P(a number bigger than 6) = 0
6
f) P(an integer) =
6
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= 1
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Calculating probability
The children in a class were asked how many siblings
(brothers and sisters) they had. The results are shown in
this frequency table:
Number of siblings
0
1
2
3
4
5
6
7
Number of pupils
4
8
9
4
3
1
0
1
What is the probability that a pupil chosen at random from
the class will have two siblings?
There are 30 pupils in the class and 9 of them have two
siblings.
9
3
So, P(two siblings) =
=
30
10
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Calculating probability
A bag contains 12 blue balls and some red balls.
The probability of drawing a blue ball at random from the
bag is
3
7
.
How many red balls are there in the bag?
12 balls represent
3
7
So, 4 balls represent
of the total.
1
7
and, 28 balls represent
of the total
7
7
of the total.
The number of red balls = 28 – 12 = 16
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The probability of an event not occurring
The following spinner is spun once:
What is the probability of it landing on the yellow sector?
1
P(yellow) =
4
What is the probability of it not landing on the yellow sector?
3
P(not yellow) =
4
If the probability of an event occurring is p then the
probability of it not occurring is 1 – p.
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The probability of an event not occurring
The probability of a factory component being faulty is 0.03.
What is the probability of a randomly chosen component not
being faulty?
P(not faulty) = 1 – 0.03 = 0.97
The probability of pulling a picture card out of a full deck of
cards is
3
13
.
What is the probability not pulling out a picture card?
3
10
P(not a picture card) = 1 –
=
13
13
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The probability of an event not occurring
The following table shows the probabilities of 4 events.
For each one work out the probability of the event not
occurring.
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Event
Probability of the
event occurring
Probability of the
event not occurring
A
3
5
2
5
B
0.77
0.23
C
9
20
11
20
D
8%
92%
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The probability of an event not occurring
There are 60 sweets in a bag.
10 are cola bottles,
20 are hearts,
1
are fried eggs,
4
the rest are teddies.
What is the probability that a sweet chosen at random
from the bag is:
5
a) Not a cola bottle P(not a cola bottle) =
6
b) Not a teddy
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45
3
P(not a teddy) =
=
60
4
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Mutually exclusive outcomes
Outcomes are mutually exclusive if
they cannot happen at the same time.
For example, when you toss a single coin either it will land on
heads or it will land on tails. There are two mutually exclusive
outcomes.
Outcome A: Head
Outcome B: Tail
When you roll a dice either it will land on an odd number or it
will land on an even number. There are two mutually exclusive
outcomes.
Outcome A: An odd number
Outcome B: An even number
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Mutually exclusive outcomes
A pupil is chosen at random from the class. Which of the
following pairs of outcomes are mutually exclusive?
Outcome A: the pupil has brown eyes.
Outcome B: the pupil has blue eyes.
These outcomes are mutually exclusive because a pupil can
either have brown eyes, blue eyes or another colour of eyes.
Outcome C: the pupil has black hair.
Outcome D: the pupil has wears glasses.
These outcomes are not mutually exclusive because a pupil
could have both black hair and wear glasses.
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Adding mutually exclusive outcomes
If two outcomes are mutually exclusive then their probabilities
can be added together to find their combined probability.
For example, a game is played with the following cards:
What is the probability that a card is a moon or a sun?
1
1
and
P(sun) =
3
3
Drawing a moon and drawing a sun are mutually exclusive
outcomes so,
1
1
2
P(moon or sun) = P(moon) + P(sun) =
+
=
3
3
3
P(moon) =
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Adding mutually exclusive outcomes
If two outcomes are mutually exclusive then their probabilities
can be added together to find their combined probability.
For example, a game is played with the following cards:
What is the probability that a card is yellow or a star?
1
1
P(yellow card) =
and
P(star) =
3
3
Drawing a yellow card and drawing a star are not mutually
exclusive outcomes because a card could be yellow and a star.
P (yellow card or star) cannot be found by adding.
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The sum of all mutually exclusive outcomes
The sum of all mutually exclusive outcomes is 1.
For example, a bag contains red counters, blue counters,
yellow counters and green counters.
P(blue) = 0.15
P(yellow) = 0.4
P(green) = 0.35
What is the probability of drawing a red counter from the
bag?
P(blue, yellow or green) = 0.15 + 0.4 + 0.35 = 0.9
P(red) = 1 – 0.9 = 0.1
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The sum of all mutually exclusive outcomes
A box contains bags of crisps. The probability of drawing out
the following flavours at random are:
2
1
P(salt and vinegar) =
P(ready salted) =
5
3
The box also contains cheese and onion crisps.
What is the probability of drawing a bag of cheese and onion
crisps at random from the box?
P(salt and vinegar or ready salted) =
2
1
6+5
11
+
=
=
5
3
15
15
11
4
P(cheese and onion) = 1 –
=
15
15
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The sum of all mutually exclusive outcomes
A box contains bags of crisps. The probability of drawing out
the following flavours at random are:
2
1
P(salt and vinegar) =
P(ready salted) =
5
3
The box also contains cheese and onion crisps.
There are 30 bags in the box. How many are there of each
flavour?
Number of salt and vinegar =
2
of 30 = 12 packets
5
1
Number of ready salted = 3 of 30 = 10 packets
4
Number of cheese and onion = 15 of 30 = 8 packets
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Contents
D4 Probability
D4.1 The language of probability
D4.2 The probability scale
D4.3 Calculating probability
D4.4 Probability diagrams
D4.5 Experimental probability
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Finding all possible outcomes of two events
Two coins are thrown. What is the probability of getting
two heads?
Before we can work out the probability of getting two heads
we need to work out the total number of equally likely
outcomes.
There are three ways to do this:
1) We can list them systematically.
Using H for heads and T for tails, the possible outcomes
are:
TH and HT are separate
TT, TH, HT, HH.
equally likely outcomes.
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Finding all possible outcomes of two events
2) We can use a two-way table.
Second coin
First
coin
H
T
H
HH
TH
T
HT
TT
From the table we see that there are four possible outcomes
one of which is two heads so,
1
P(HH) =
4
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Finding all possible outcomes of two events
3) We can use a probability tree diagram.
Outcomes
Second coin
First coin
H
HH
T
H
HT
TH
T
TT
H
T
Again we see that there are four possible outcomes so,
1
P(HH) =
4
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Finding the sample space
A red dice and a blue dice are thrown and their scores
are added together.
What is the probability of getting a total of 8 from both
dice?
There are several ways to get a total of 8 by adding the
scores from two dice.
We could get a 2 and a 6, a 3 and a 5, a 4 and a 4,
a 5 and a 3, and a 6 and a 2.
To find the set of all possible outcomes, the sample
space, we can use a two-way table.
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Finding the sample space
+
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2
3
4
5
6
7
3
4
5
6
7
8
4
5
6
7
8
9
5
6
7
8
9
10
6
7
8
9
10
11
7
8
9
10
11
12
From the sample
space we can see
that there are 36
possible outcomes
when two dice are
thrown.
Five of these have
a total of 8.
5
P(8) =
36
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Scissors, paper, stone
In the game scissors, paper, stone two players have to show
either scissors, paper, or stone using their hands as follows:
scissors
paper
stone
The rules of the game are that:
Scissors beats paper (it cuts).
Paper beats stone (it wraps).
Stone beats scissors (it blunts).
If both players show the same hands it is a draw.
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Scissors, paper, stone
What is the probability that both players will show the
same hands in a game of scissors, paper, stone?
We can list all the possible outcomes in a two-way table using
S for Scissors, P for Paper and T for sTone.
Second player
First
player
Scissors
Paper
Stone
Scissors
SS
SP
ST
Paper
PS
PP
PT
Stone
TS
TP
TT
3
1
P(same hands) =
=
9
3
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Scissors, paper, stone
What is the probability that the first player
will win a game of scissors, paper, stone?
Using the two-way table we can identify all the ways that the
first player can win.
Second player
First
player
Scissors
Paper
Stone
Scissors
SS
SP
ST
Paper
PS
PP
PT
Stone
TS
TP
TT
3
1
P(first player wins) =
=
9
3
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Scissors, paper, stone
What is the probability that the second player
will win a game of scissors, paper, stone?
Using the two-way table we can identify all the ways that the
second player can win.
Second player
First
player
Scissors
Paper
Stone
Scissors
SS
SP
ST
Paper
PS
PP
PT
Stone
TS
TP
TT
3
1
P(second player wins) =
=
9
3
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Scissors, paper, stone
Is scissors, paper, stone a fair game?
P(first player wins) =
1
3
1
P(second player wins) =
3
1
P(a draw) =
3
Both player are equally likely to win so, yes, it is a fair game.
Play scissors paper stone 30 times with a partner.
Record the number of wins for each player and the
number of draws. Are the results as you expected?
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Contents
D4 Probability
D4.1 The language of probability
D4.2 The probability scale
D4.3 Calculating probability
D4.4 Probability diagrams
D4.5 Experimental probability
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Estimating probabilities based on data
What is the probability a person chosen
at random being left-handed?
Although there are two possible outcomes, right-handed and
left-handed, the probability of someone being left-handed is
not ½, why?
The two outcomes, being left-handed and being righthanded, are not equally likely. There are more right-handed
people than left-handed.
To work out the probability of being left-handed we could
carry out a survey on a large group of people.
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Estimating probabilities based on data
Suppose 1000 people were ask whether they were
left- or right-handed.
Of the 1000 people asked 87 said that they were lefthanded.
From this we can estimate the probability of someone being
left-handed as
87
1000
or 0.087.
If we repeated the survey with a different sample the results
would probably be slightly different.
The more people we asked, however, the more accurate our
estimate of the probability would be.
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Relative frequency
The probability of an event based on data from an experiment
or survey is called the relative frequency.
Relative frequency is calculated using the formula:
Number of successful trials
Relative frequency =
Total number of trials
For example, Ben wants to estimate the probability that a
piece of toast will land butter-side-down.
He drops a piece of toast 100 times and observes that it
lands butter-side-down 65 times.
65
13
Relative frequency =
=
100
20
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Relative frequency
Sita wants to know if her dice is fair. She throws it 200 times
and records her results in a table:
Number Frequency Relative frequency
1
31
2
27
3
38
4
30
5
42
6
32
31
200
27
200
38
200
30
200
42
200
32
200
= 0.155
= 0.145
= 0.190
= 0.150
= 0.210
= 0.160
Is the dice fair?
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Expected frequency
The theoretical probability of an event is its calculated
probability based on equally likely outcomes.
If the theoretical probability of an event can be calculated,
then when we do an experiment we can work out the
expected frequency.
Expected frequency = theoretical probability × number of trials
For example, if you rolled a dice 300 times, how many times
would you expect to get a 5?
The theoretical probability of getting a 5 is
1
So, expected frequency =
× 300 = 50
6
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1
6
.
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Expected frequency
If you tossed a coin 250 times how many
times would you expect to get a tail?
1
Expected frequency =
× 250 = 125
2
If you rolled a fair dice 150 times
how many times would you expect
to a number greater than 2?
2
Expected frequency =
× 150 = 100
3
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Spinners experiment
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Random results
Remember that when an experiment is carried out the results
will be random and unpredictable.
Each time the experiment is repeated the results will be
different.
The more times an experiment is repeated the more accurate
the estimated probability will be.
Jenny throws a dice 12 times and doesn’t get a six. She
concludes that the dice must be biased.
Although you would expect to get two sixes in twelve
throws it is possible that you won’t. You would have to
thrown the dice many more times to find out if it is biased.
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