Bartholomew Maths

Data 2 Lesson 7 to
Lesson 10
Name: ______________________
Teacher: ______________________
 Data 2

The Probability Scale

You may often hear people making comments such as ‘my team is certain to win’. Words
like ‘certain’, ‘impossible’, ‘likely’, ‘unlikely’ and ‘evens’ can be used to describe the chance
of an event happening. Here is a likelihood scale featuring these words.

Write down the most suitable word(s) from the likelihood scale above to describe the
probability of each of these events and mark the likelihood of each event on the scale with
its corresponding letter.

A It will snow in London in December.

B If you roll a dice you will get a number bigger than three.

C A baby will be born in Bangkok today.

D A cow will turn into a horse.

E Manchester United will win more games than they draw.

F A fair coin will fall showing heads.

G If you drop a drawing pin, it will land point up.

H A randomly picked playing card will be a picture card.

I If there are 5 people in a bus queue the oldest will be at the back.

J Someone will win the jackpot in the next lottery draw.

K Tomorrow Facebook will be mentioned on BBC TV.

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 Data 2

The Probability Scale

Probability is a numerical measure of the chance of an event happening.

A probability of 0 means it is impossible for the event to happen.
A probability of 1 means the event is certain to happen.

Write down a number between 0 and 1 (as a decimal) to represent the probability of each
of these events and mark the likelihood of each event on the scale with its corresponding
letter. You are not expected to calculate the probabilities.

A A baby born will be a boy.

B There will be a hurricane in Britain in the next week.

C You will score ten when a normal six-diced dice is rolled.

D A person chosen at random was born on a weekday.

E There will be four aces in a complete pack of 52 playing cards.

F It will rain on at least one day in August next year.

G Everyone in Great Britain will make at least one phone call today.

Describe an event that you think has a probability of:

1 0.3
2 1
3 0.8

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 Data 2

Venn Diagrams (Frequency)

Place the following numbers correctly in Place the following numbers correctly in the
the diagram below. diagram below.

5, 1, 12, 9, 6, 18, 3, 10, 20, 2, 36, 15 15, 5, 3, 10, 30, 1, 12, 4, 9, 2, 45, 20

Work out how many of the numbers are: Work out how many of the numbers are:

factors of 18 factors of 20

factors of 30 factors of 45

factors of both 18 and 30 factors of both 20 and 45

This Venn diagram shows information

Work out how many of the students:
about some students in a particular class.

wear glasses
M G
are male
8 3 2
do not wear glasses

7 are not male

are male and wear glasses

M is the event the student is male are male and do not wear glasses

are not male and do not wear

G is the event the student wears glasses glasses

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Venn Diagrams (Probability)

Place the following numbers correctly in Place the following numbers correctly in the
the diagram below. diagram below.

5, 1, 12, 9, 6, 18, 3, 10, 20, 2, 36, 15 15, 5, 3, 10, 30, 1, 12, 4, 9, 2, 45, 20

A number is selected at random, work out A number is selected at random, work out the
the probability the number is: probability the number is:

a factor of 18 a factor of 20

a factor of 30 a factor of 45

a factor of both 18 and 30 a factor of both 20 and 45

This Venn diagram shows information A student is selected at random, work out the
about some students in a particular class. probability that the student:

wears glasses
M G
is male
8 3 2
does not wear glasses

7 is not male

is male and wears glasses

M is the event the student is male is male and does not wear glasses

G is the event the student wears glasses is male given they wear glasses

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 Data 2

Equally Likely Outcomes (Coins)

Outcomes are the possible results of a probability experiment.

The two possible outcomes when you flip a coin are heads (H) and tails (T).
If the coin is fair, then these two outcomes are equally likely.

Write down all the possible outcomes of flipping two coins.

1

Write down all the possible outcomes of flipping three coins.

2

Write down all the possible outcomes of flipping four coins.

Complete this table for the total number of possible outcomes when different
numbers of coins are flipped together.

Number of Coins Total Number of Possible Outcomes

1 2

4 3

10

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 Data 2

Equally Likely Outcomes (Dice)

The six possible outcomes when you roll a dice are 1, 2, 3, 4, 5 and 6.
If the dice is fair, then these six outcomes are equally likely.
Complete the two-way table to show all the outcomes when two dice are rolled and
their scores are added.
Work out the probability of
a
scoring a total of 7.
Work out the probability of
b
scoring a total of 8.
Work out the probability of
1 c
scoring a total of 12.
Work out the probability of
d
scoring a total of at least 5.
Work out the probability of
e
scoring a total of more than 10.
Work out the probability that the
f
total score is odd.
Complete the two-way table to show all the outcomes when two dice are rolled and
their scores are multiplied.
Work out the probability of
a
scoring 18.
Work out the probability of
b
scoring 1.
Work out the probability of
2 c
scoring at least 20.
Work out the probability of
d
scoring less than 9.
Work out the probability of
e
scoring at least 7.
Work out the probability that the
f
score is even.

Two tetrahedral dice

are rolled and their
3* scores are added.
Write down all the
possible outcomes.

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Equally Likely Outcomes (Playing Cards)

There are 52 playing cards in a standard pack (not including jokers). There are four suits:
hearts, clubs, diamonds and spades. Each suit is made up of 13 cards: Ace, 2, 3, 4, 5, 6, 7,
8, 9, 10, Jack, Queen and King. The Jack, Queen and King are known as picture cards. The
hearts and diamonds are red cards and the clubs and spades are black cards.
If a card is drawn at random and each card is equally likely to be drawn, work out the
probability that the card selected is:

1 a red card 16 not a heart

2 a club 17 not a black card

3 a picture card 18 not a King

4 an ace 19 not a picture card

5 a black card 20 not a Jack

6 a two 21 not an ace

7 a Queen 22 not a red five

8 a card with an even number 23 not the King of clubs

9 a black picture card 24 not a joker

10 a red ace A brief history of the playing cards we use today…

The deck of 52 French playing cards is the most common

11 a black ten deck of playing cards used today and widely used across
the United Kingdom.
12 a black card with an odd number Playing cards arrived in Europe from Egypt around 1370
and were already reportedly being used in France in 1377.
One of the most distinguishing features of the French cards
13 a red card without a picture is the Queen; the original Egyptian cards and their
derivatives, the Latin suited and German suited cards, all
14 the seven of hearts have three male face cards.
In the 19th century, corner indices and rounded corners
were added and cards became reversible, relieving players
15 the ace of spades from having to flip face cards right side up.

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 Data 2

Theoretical Probability
What is the probability of the spinner
1
landing on a 3?
What is the probability of the spinner
2
landing on a 1?
What is the probability of the spinner
3
landing on a 1 or a 3?
Are you more likely to spin an odd
4
number or an even number?
What is the probability of the spinner
5
landing on red?
What is the probability of the spinner
6
landing on orange?
What is the probability of the spinner
7
landing on red or blue?
What is the probability of the spinner
8
landing on a colour other than red?

Tom says, ‘You have a fifty-fifty

9 chance of spinning red.’
Explain what he means.

Sarah says that because there are

10 three options the probability of
1
getting blue is . Do you agree?
3

Here are some marbles; they are either

grey, white or black.
The marbles are placed in a bag and one is
drawn at random.
What is the probability of drawing Would you be more likely to draw a
11
the black marble? marble that is not black or a marble that
15 is not grey? Explain your answer.
What is the probability of drawing a
12
grey marble?
What is the probability of the If three more black marbles were added
13
drawing a white marble? to the bag, what would be the
16 probability of drawing a black marble?
What is the probability of drawing a
14
marble that is not white?

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 Data 2

Equally Likely or Not Equally Likely

Decide whether each of these represent equally likely outcomes or not.

1 Flipping a coin and obtaining a head or a tail. YES / NO

2 Passing or failing a driving test. YES / NO

3 Rolling a fair dice and obtaining an even or an odd number. YES / NO

Picking a red ball or a blue ball from a bag of mixed red and blue
4 YES / NO
balls.
Rolling a double and not rolling a double when two fair dice are
5 YES / NO
rolled together.

6 Catching or missing a train. YES / NO

7 Raining or not raining tomorrow. YES / NO

Theoretical or Not Theoretical

Decide whether each of these can be worked out theoretically or not.

1 The probability that a drawing pin will land point down. YES / NO

2 The probability of winning the UK National Lottery. YES / NO

3 The probability of snow on the 25th December 2018. YES / NO

4 The probability of obtaining a head when a fair coin is flipped. YES / NO

5 The probability of rolling a six with a weighted six-sided dice. YES / NO

The probability that two people in the same class share the same
6 YES / NO
birthday.

7 The probability that you achieve at least 50% in the Data 2 Test. YES / NO

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 Data 2

Total Probability

For each event, there are a set of outcomes. Each outcome has an associated probability.
The sum of these probabilities is always 1, as obtaining at least one of these outcomes is
certain. Work out the missing numbers in these calculations that all sum to one.

1
1 0.5 + = 1 17 2
+ = 1

1
2 0.2 + = 1 18 4
+ = 1

1
3 0.7 + = 1 19 + 3
= 1

1
4 + 0.9 = 1 20 + 5
= 1

2
5 + 0.4 = 1 21 7
+ = 1

3
6 0.25 + = 1 22 10
+ = 1

8
7 0.85 + = 1 23 9
+ = 1

5
8 + 0.43 = 1 24 + 6
= 1

4
9 + 0.07 = 1 25 + 13
= 1

7
10 + 0.64 = 1 26 12
+ = 1

1 1
11 0.475 + = 1 27 2
+ 4
+ = 1

1 1
12 0.291 + = 1 28 + 3
+ 6
= 1

3 1
13 0.723 + = 1 29 8
+ + 4
= 1

1 3
14 + 0.518 = 1 30 2
+ 10
+ = 1

1 2
15 + 0.682 = 1 31 + 4
+ 5
= 1

5 3
16 + 0.999 = 1 32 12
+ + 8
= 1

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 Data 2

Frequency Trees 1

There are 30 pupils in a class.

There are 14 boys in the class.

1 Complete the frequency tree.

Find the probability of a pupil chosen

at random being a girl.

There are 28 pupils in a class.

There are 18 boys and 10 girls in the
class. Of the boys, 3 wear glasses.
Of the girls, 8 do not wear glasses.

2
Complete the frequency tree.

Find the probability of a pupil chosen

at random being a girl who wears
glasses.

There are 70 members in a

badminton club.
There are 45 males in the club.
Of the males, 10 are left-handed.
Of the females, 18 are right-handed.
3
Complete the frequency tree.

Find the probability of a player

chosen at random being a right-
handed male.

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 Data 2

Frequency Trees 2
There are 140 pupils in a year group.
There are 60 males in the year
group.
Of the males, 24 have a pet.
Of the females, 65 do not have a
pet.
1
Complete the frequency tree.

Find the probability of a pupil

chosen at random being a female
who has a pet.

There are 140 pupils in Year 7.

75 are male, of these 14 were late to
school on Monday.
There were 8 girls who were late to
school on Monday.
2
Complete the frequency tree.

Use your frequency tree to work out

the probability of a student chosen
at random being a male who is not
late for school on Monday.

Year 9 has 250 students.

There are 2 bands in year 9: X band
and Z band. There are 140 pupils in
X band, of these 80 are female.
There are 65 males in Z band.
3
Complete the frequency tree.

Use your frequency tree to work out

the probability of a pupil chosen at
random being a female from Z band.

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 Data 2

Frequency Trees 3

There are 96 members of a running

club. 52 of the members are male,
the rest are female.

1
4
of the female members are under
1 21, the rest are 21 or over.

75% of the male members are under

21, the rest are 21 or over.

Use your completed frequency tree to work out the probability of a member chosen
at random being a female aged 21 or over.
There are 300 pupils in Year 11.
There are 2 bands in year 11: X band
and Z band.

There are 138 pupils in X band.

2
The ratio of males to females in the X
band is 5:1
The ratio of males to females in the Z
band is 7:2
Use your completed frequency tree to work out the probability of a pupil chosen at
random being a male from X band.
73 people took a test. Before the
test, they predicted whether they
would pass or fail.

50 people predicted they would pass.

3
55 people did pass. Of these 55
people, four times as many people
predicted they would pass as
predicted they would fail.

Draw a frequency tree to represent this information.

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 Data 2

The ‘Bartholomew’ Grand National

This is a whole class activity and your teacher will explain the rules to you. Good luck!

START
1 2 3 4 5 6 7 8 9 10 11 12

FINISH

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 Data 2

Combined Events
Represent all the possible outcomes of rolling a fair six-sided dice and flipping a fair
coin in the table below.

1 2 3 4 5 6
1 H H, 1
T
Use your completed table to work out the probability that you obtain and tail
and an even number when you roll a fair six-sided dice and flip a fair coin.

John has an equal number of shirts in blue, white and green.

John has only two pairs of trousers, one pair is blue and one pair is purple.
Represent all the possible outcomes in the table below.

Use your completed table to work out the probability that John randomly
picks a green shirt and a purple trousers to wear.

These two spinners are spun and their

scores are added. Construct a table in your
exercise books which represents the all the
possible outcomes.

Use your completed table to work out the

3
probability that you:

get a total of 10 get a total of less than 9

get a total of 6 get a total of more than 5

get a total of greater than or

get an even total
equal to 7

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 Data 2

Experimental Probability 1

A weighted coin has been flipped a Outcome Frequency

number of times and the results have
been summarised in a table opposite. Head 4

Using only the results recorded in this

1 Tail 6
table work out the probability that
the next time the coin is flipped you
Total
get:

a head a tail

The same coin is flipped another 40

Outcome Frequency
times and the results have been
added to the ones above and
Head 19
summarised in the table opposite.
Using only the results recorded in this
2 Tail 31
table work out the probability that
the next time the coin is flipped you
Total
get:

a head a tail

The same coin is flipped a further 150

Outcome Frequency
times and the results have been
added to the ones above and
Head 74
summarised in the table opposite.
Using only the results recorded in this
3 Tail 126
table work out the probability that
the next time the coin is flipped you
Total
get:

a head a tail

Using the information provided in the tables, how would you describe this coin.
4

Conduct a similar experiment to investigate the likelihood of dropping a drawing pin

Ext
and it landing point down.

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 Data 2

Experimental Probability 2
A weighted six-sided dice has been
Outcome Frequency
rolled a number of times and the
results have been summarised in a
1 2
table opposite.
Using only the results recorded in this
2 7
table work out the probability that
the next time the dice is rolled you
3 5
get:
1
a five 4 4

a three 5 3

a number less than 4 6 9

a number more than 5 Total

The same dice is rolled another 150

Outcome Frequency
times and the results have been
added to the ones above and
1 12
summarised in the table opposite.
Using only the results recorded in this
2 27
table work out the probability that
the next time the dice is rolled you
3 34
get:
2
a five 4 32

a three 5 24

a number less than 4 6 51

a number more than 5 Total

Using the information provided in both tables, how would you describe this dice.
3

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 Data 2

Expected Outcomes
1
The probability of winning a particular game is
4

Tommy plays the game 12 times.

How many times is he likely to win?
1
Timmy plays the same game 36 times.
How many times is he likely to win?
Tammy plays the same game 60 times.
How many times is she likely to lose?
To win a small prize you must spin a spinner and land in the prize sector.
The probability of landing in the sector labelled ‘prize’ is 0.3
Danny spins the spinner 10 times.
How many times is he likely to win?
2
Donny spins the spinner 40 times.
How many times is he likely to win?
Debbie spins the spinner 80 times.
How many times is she likely to lose?
To win a prize you must pick a white ball, out of mixed box of black and white balls.
There are 5 balls in the box and of these 2 are white. The balls are always replaced
are being picked. If you only play the game once, what is the probability of winning
the prize?
Jackie plays the game 20 times.
3
How many times is she likely to win?
Jimmy plays the game 35 times.
How many times is he likely to win?
Jamie plays the game 90 times.
How many times is he likely to lose?
2
The probability of losing a different game is
3

Robbie plays the game 18 times.

How many times is he likely to lose?
Ronnie plays the game 54 times.
4*
How many times is he likely to win?
Reggie plays the game 99 times.
How many times is he likely to win?
Ruthie plays the game only 5 times.
How many times is she likely to win?

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 Data 2

Are you ready? Data 2 Next steps …

 Do you know simple fraction, decimal and  Know the meaning of words such as likelihood,  Be able to list outcomes of combined events using
percentage equivalences (for example: for 10%, probability, impossible, certain and even chance a probability tree diagram
20%, 25%, 50%, 75%, 100%)?  Be able to place events on a 0-1 probability scale  Be able to label a tree diagram with probabilities
 Can you add fractions with different  Be able to complete Venn diagrams when events are independent
denominators?  Be able to calculate probabilities from Venn  Be able to label a tree diagram with probabilities
 Do you know the four different suits that are diagrams when events are dependent
found in a standard pack of playing cards?  Be able to write down outcomes systematically in  Know that relative frequency and experimental
lists or on two-way probability space diagrams probability are the same thing
 Be able to calculate theoretical probabilities from
lists of outcomes or two-way probability space
diagrams
 Be able to use the fact that the probability of all
outcomes adds up to 1
 Be able to complete frequency trees and make
conclusions about probabilities based on
frequency trees
 Be able to construct theoretical possibility spaces
for two combined events
 Be able to calculate experimental probabilities
from a frequency table
 Be able to use theoretical and experimental
probabilities to calculate expected frequencies

Glossary
The chance that a particular outcome will occur, measured as a ratio of the total of
Probability possible outcomes. Two-way table Two-way tables are used to study the relationship between categorical variables.

A number between 0 and 1 that indicates the chance or likelihood of an event

Likelihood The chance that a particular outcome will occur. Theoretical probability happening.

Venn diagram A diagram using circles or other shapes, to show the relationship between sets. Probability experiment A situation where a number of trials are conducted to determine probability.

Number of times a specific outcome is expected to occur when a probability

Outcome Result of one trial in a probability experiment. Expected frequency experiment is repeated a number of times.

Equally likely outcomes Outcomes with the same probability. Observed frequency Number of times a specific outcome occurred.