Probability
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
even
chance
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
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?
bag c
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 gets a point, if it
doesn’t team B gets 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.
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 gets a point.
If a square number is drawn team B gets a point.
If any other number is drawn team C gets 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.
Fair games
A spinner has five equal sectors numbered 1 to 5.
The spinner is spun many times.
If the spinner stops on an even
5
number team A gets 3 points.
If the spinner stops on an odd
4
number team B gets 2 points.
1
2
3
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.
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 no
win win
no
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?
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?
Probability statements
Statements involving probability are often incorrect or misleading. Discuss the
following statements:
The number 18 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.
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.
Contents
D4 Probability
D D4.1 The language of probability
2
D D4.2
2
The probability scale
D D4.3 Calculating probability
2
D D4.4 Probability diagrams
2
D D4.5 Experimental probability
2
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
More likely
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.
Listing possible outcomes
When you roll a fair dice you are equally likely to get one of six possible
outcomes:
1
1
1
1
1
1
6
6
6
6
6
6
Since each number on the dice is equally likely the probability of getting any
one of the numbers is 1 divided by 6 or
.
1
6
Calculating probability
What is the probability of the following events?
1) A coin landing tails up?
P(tails) =
3) Drawing a seven of hearts
from a pack of 52 cards?
1
P(7 of
2
2) This spinner stopping on
the red section?
P(red) =
)=
1
52
4) A baby being born on a
Friday?
1
4
P(Friday) =
1
7
Calculating probability
If the outcomes of an event are equally likely then we can calculate the
probability using the formula:
Number of successful outcomes
Probability of an event =
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?
P(green) =
3
10
or 0.3
or 30%
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?
a) P(red) =
b) P(blue) =
c) P(green) =
2
8
1
=
4
1
8
4
8
=
1
2
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
b) P(a multiple of 3) =
c) P(an odd number) =
2
1
=
6
3
6
3
=
1
2
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?
3
d) P(a prime number) =
6
e) P(a number bigger than 6) =
f) P(an integer) =
6
6
= 1
=
1
2
0
Don’t write
0
6
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.
So, P(two siblings) =
9
30
=
3
10
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
of the total.
7
So, 4 balls represent
1
of the total
7
and, 28 balls represent
The number of red balls = 28 – 12 =
7
of the total.
7
16
The probability of an event not occurring
The following spinner is spun once:
What is the probability of it landing on the yellow sector?
P(yellow) =
1
4
What is the probability of it not landing on the yellow sector?
P(not yellow) =
3
4
If the probability of an event occurring is p then the probability of it
not occurring is 1 – p.
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
.3
13
What is the probability
of not pulling out a picture card?
cards is
P(not a picture card) = 1 –
=
3
10
13
13
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.
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%
Event
The probability of an event not occurring
There are 60 sweets in a bag.
10 are cola bottles,
1
4
20 are hearts,
are fried eggs,
the rest are teddies.
What is the probability that a sweet chosen at random from the bag is:
a) Not a cola bottle
b) Not a teddy
5
P(not a cola bottle) =
P(not a teddy) =
6
45
60
=
3
4
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
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.
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?
P(moon) =
1
and
P(sun) =
1
3
3
Drawing a moon and drawing a sun are mutually exclusive outcomes so,
P(moon or sun) = P(moon) + P(sun) =
1
3
+
1
3
=
2
3
Adding mutually exclusive outcomes
What is the probability that a card is yellow or a star?
P(yellow card) =
1
1
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 simply by adding.
We have to subtract the probability of getting a yellow star.
P(yellow card or star) =
1
3
+
1
3
–
1
9
=
3+3–1
9
=
5
9
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 or yellow or green) = 0.15 + 0.4 + 0.35 =
P(red) = 1 – 0.9 =
0.1
0.9
The sum of all mutually exclusive outcomes
A box contains bags of crisps. The probability of drawing out the following flavours
at random are:
P(salt and vinegar) =
2
5
1
P(ready salted) =
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?
2
P(salt and vinegar or ready salted) =
P(cheese and onion) = 1 –
5
11
15
=
4
15
+
1
3
=
6+5
15
=
11
15
The sum of all mutually exclusive outcomes
A box contains bags of crisps. The probability of drawing out the following flavours
at random are:
P(salt and vinegar) =
2
5
P(ready salted) =
1
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 =
Number of ready salted =
Number of cheese and onion =
2 of 30 =
5
12 packets
1 of 30 = 10 packets
3
4 of 30 = 8 packets
15
Contents
D4 Probability
D D4.1 The language of probability
2
D D4.2 The probability scale
2
D D4.3 Calculating probability
2
D D4.4
2
Probability diagrams
D D4.5 Experimental probability
2
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:
TT,
TH,
HT,
HH.
TH and HT are separate equally
likely outcomes.
Finding all possible outcomes of two events
2) We can use a two-way table.
Second coin
First
coin
H
T
H
HH
HT
T
TH
TT
From the table we see that there are four possible outcomes one of which is two
heads so,
P(HH) =
1
4
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
T
TT
H
TH
T
Again we see that there are four possible outcomes so,
P(HH) =
1
4
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 5 and a 3,
a 3 and a 5,
a 4 and a 4,
or a 6 and a 2.
To find the set of all possible outcomes, the sample space, we can use a
two-way table.
Finding the sample space
+
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
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.
P(8) =
7
8
9
10
11
12
5
36
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
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.
stone
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
P(same hands) =
3
9
=
1
3
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
P(first player wins) =
3
9
=
1
3
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
P(second player wins) =
3
9
=
1
3
Scissors, paper, stone
Is scissors, paper, stone a fair game?
P(first player wins) =
P(second player wins) =
P(a draw) =
1
3
1
3
1
3
Both players 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?
Contents
D4 Probability
D D4.1 The language of probability
2
D D4.2 The probability scale
2
D D4.3 Calculating probability
2
D D4.4 Probability diagrams
2
D D4.5
2
Experimental probability
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 right-handed, 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.
Estimating probabilities based on data
Suppose 1000 people were asked whether they were left- or right-handed.
Of the 1000 people asked 87 said that they were left-handed.
From this we can estimate the probability of someone being
left-handed as
87 or 0.087.
1000
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.
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.
Relative frequency =
65
100
=
13
20
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
1
31
2
27
3
38
4
30
5
42
6
32
Relative frequency
31
200
27
200
38
200
30
200
42
200
32
200
Is the dice fair?
= 0.155
= 0.135
= 0.190
= 0.150
= 0.210
= 0.160
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
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
So, expected frequency =
1
× 300 =
6
. 1
6
50
Expected frequency
If you tossed a coin 250 times how many times would
you expect to get a tail?
Expected frequency =
1
× 250 =
2
125
If you rolled a fair dice 150 times how many
times would you expect to a number greater
than 2?
Expected frequency =
2
× 150 =
3
100
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 throw the dice many more times to find out if it is
biased.