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Key words
Probability using words
event
probability scale
random
Describe probability using words
You can describe the probability or (chance) of an event happening in words. This
probability scale shows some of them.
0
1
impossible
even
certain
A random event is one that can’t be predicted. Throwing a dice is one way of
producing random numbers.
Example 1
Jack chooses a counter at random from this jar.
Draw a probability scale showing the approximate
chance of each colour being chosen.
0
1
green
red
blue
Example 2
a) Colour this spinner so it is less likely that it will land on
blue than yellow.
b) Colour this spinner so there is an equal chance of landing
on green and yellow but it is most likely to land on blue.
a)
or
b)
Exercise 3.1
Here is the net of a dice.
a)
b)
c)
d)
26
Which number is most likely to be thrown?
Which number is least likely to be thrown?
Which number has an even chance of being thrown?
It is impossible to throw some numbers between 1
and 6 using this dice. Which ones are they?
Maths Connect 2G
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50 raffle tickets are sold, numbered from 1 to 50. A ticket is chosen at random, so each
has the same chance of being chosen.
Match each event to the chance of it being chosen. One has already been done for you.
A
B
C
D
E
The number 16
The number 75
An odd number
A number less than 51
A number bigger than 10
1
2
3
4
5
certain
likely
impossible
unlikely
even
A counter is chosen at random from each jar.
E2 (It is likely that a number
bigger than 10 will be chosen.)
a)
b)
Draw a probability scale for each jar to show
the approximate chance of a yellow, blue, red
and green counter being chosen.
Use Example 1 to help you.
You need blue, green and red colouring pencils. Copy this spinner twice.
a) Colour the spinner so it is impossible for it to land
on blue and more likely to land on green than red.
b) Colour the spinner so it is most likely to land on
blue but green and red have an equal chance as each other.
Here are the nets of three different dice.
Dice 1
Dice 2
Dice 3
Which dice would you choose so:
a) there is an equal chance of throwing an odd or even number
b) it is certain to throw a number less than 5
c) there is an equal chance of throwing a 5 or 6
d) it is impossible to throw a 3?
There are two answers
to this question.
Eliza says that she will either meet a zebra on her way to school or she will not, so there
is an even chance of meeting a zebra. Is Eliza right? Explain your answer.
Investigation
Think of an event which is:
impossible
b) certain
has an even chance of happening
has a chance between even and certain of happening
has a chance between impossible and
even of happening.
0
Place each of your events on the probability scale. impossible
a)
c)
d)
e)
1
even
certain
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Key words
Probability using numbers
random
theoretical
Describe probability using fractions, decimals and percentages
A probability can be written as a fraction or decimal. This value must be from 0 to 1.
A probability can also be written as a percentage.
A random choice is one that can’t be predicted. The theoretical
probability of choosing a red counter at random is 25. This is because
there are two red counters out of a total of five, and they each have the
same chance of being chosen.
2
5
0.4 40%, so the probability can also be written as 0.4 or 40%.
This probability should not be written as ‘2 in 5 chance’, ‘2 : 5’ or ‘2 out of 5’.
Example
A bag contains ten Scrabble tiles:
t
t
u
t
u
t
o
t
t
u
A tile is taken out at random. Find the probability that the letter is a:
a) ‘t’
b) ‘o’
c) ‘u’
d) a vowel
e) ‘i’
f) ‘t’, ‘o’ or ‘u’
Give each answer as a fraction, decimal and percentage.
Draw a probability scale for each of these events.
a)
6
10
0.6 60%
b)
1
10
0.1 10%
c)
3
10
0.3 30%
d)
4
10
0.4 40%
e) 0
f)
10
10
1 100%
5
10
0
i
o
u vowel
1
t
t, u or o
Exercise 3.2
Sam has ten pens in his pencil case. Three are blue, two are red and five are black. He
takes a pen out at random. What is the probability that the pen is:
a) black
b) blue
d) green
e) black, blue or red?
c) red
f) Draw a probability scale showing each of these events.
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Maths Connect 2G
Use the Example to help you.
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A dice is made from this net.
The dice is then thrown.
a) What is the probability that the dice shows:
i) 4
ii) 3
iii) 1 or 4
iv) an even number
v) 6
vi) 1, 3 or 4?
b) Copy and complete the probability scale to show your answers.
3
6
0
1
In a class there are 14 boys and 11 girls. The teacher chooses a pupil at random to hand
out the exercise books. What is the probability that the teacher chooses a boy?
A school sells 100 raffle tickets. Tickets numbered from 1 to 60 are blue and from 61 to 100
are red. A ticket is drawn at random. What is the probability that this ticket is:
a) blue
b) red
c) an even number
d) a number that ends in zero
e) the number 23
f) blue and a number greater than 60.
Draw a spinner that could be described by this probability scale.
0
0.25
0.5
green
red
blue
0.75
1
Ruby has 4 sweets in a bag. She always chooses a sweet without
1.
2
looking. The probability of her choosing a mint is Ruby eats one of
the mint sweets. What is the probability that the next sweet she
chooses at random is also a mint?
You could use four
counters to help.
Mrs Smith has five keys in her bag. She chooses a key at random.
a) What is the probability that she chooses the key that she needs?
b) The key Mrs Smith chooses does not open her door. What is the probability that the
next key she chooses opens her door if:
i) she puts the key she has already used back in her bag first
ii) she does not put the key she has already used back in her bag.
You can try this out using five different coloured counters.
Mrs Smith also has a pair of red gloves and a pair of blue gloves in
You could use two
red and two blue
counters to help.
her bag.
a) What is the probability that the first glove she chooses at
random is blue?
b) What is the greatest number of gloves she needs to take out from her bag to give a
pair of the same colour?
Investigation
Use the idea from Q5 to draw three different spinners and their probability scales.
Use fractions rather than decimals for your probability scale.
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Key words
Possible outcomes
outcome
event
sample space
diagram
Record outcomes using diagrams and tables
You can show possible outcomes of an event using a table or a sample space diagram .
For example, for the event ‘tossing a coin’, the outcome is either a tail or a head.
Example 1
Find all possible outcomes when two coins are tossed.
Coin 1
Head
Tail
Head
Head, head
Tail, head
Tail
Head, tail
Tail, tail
There are four possible outcomes.
Coin 2
Example 2
Ali, Ben and Colin go to the fair. Only two of the boys can go on a ride at the
same time.
a) What are all the possible outcomes?
b) What is the probability that Ali and Colin go on the ride together?
a) Ali and Ben; Ali and Colin; Ben and Colin.
b)
There are three possible outcomes
and only one of them is Ali and
Colin.
1
3
Exercise 3.3
A shop sells chocolate, vanilla or strawberry ice cream. Sangheeta would like to try
two different flavours. What combinations could she have?
Another shop sells ice creams and a choice of
topping.
a) Mark wants one flavour of ice cream with a topping.
What are his choices?
b) Louise wants two different flavours of
ice cream and a topping. What are her choices?
Ice cream
Mint
Vanilla
Banana
Topping
Sauce
or
Flake
Copy and complete the table to show all the possible outcomes when a dice is thrown
and a coin is spun.
Dice
Coin
Head
1
2
3
1, head
Tail
What is the probability of getting 4, tail?
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Maths Connect 2G
4
5
6
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Luke has number cards 1, 3, 5. Adam has number cards 2, 4, 6.
They each choose a card at random and add the two numbers
together. Copy and complete the table to show all of the
possible outcomes.
1
2
3
3
5
3
5
4
6
a) Use your table to find the probability that the total will be:
i) 5
ii) less than 6
iii) greater than 6
iv) an even number.
b) Complete the table to show all of the possible outcomes
if the numbers are multiplied together. Use your table to
find the probability that the total will be:
i) 5
ii) less than 6
iii) greater than 6
iv) an even number.
Are your answers the same for each table?
1
2
2
4
6
5
Pran spins these two spinners. He adds together the
1
2
8
4
numbers that each spinner lands on. Draw a table to
show all of the different possible totals.
3
Is the probability of an even total the same as the
probability of an odd total? Explain your answer.
6
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Matt has the number cards 4, 5 and 6.
64 is an example of a 2-digit number.
a) Which 2-digit numbers can he make?
b) What is the probability that a 2-digit number chosen at random is:
i) even
ii) less than 50
iii) not less than 50
iv) divisible by 5?
Investigation
Copy and complete these two tables showing the outcomes when two dice are thrown.
a) Fill in the boxes by adding the numbers on the dice.
1
2
1
2
3
3
4
5
6
2
3
7
4
5
6
b) Fill in the boxes by subtracting the smallest number from the largest.
1
1
0
2
2
3
4
5
6
1
You will need these results
in your next lesson.
3
4
2
5
6
4
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Key words
Tallies and frequency tables
tally
frequency
Collect data from a simple experiment and record it in a frequency table
A frequency table is a way of sorting data into groups. It is often quicker and easier to
record data using a frequency table.
Example
Eric asks some pupils in his school how many pieces of homework they got
yesterday.
0
1
1
1
2
1
1
4
1
2
2
1
1
1
3
1
2
1
3
1
0
1
0
2
Draw a tally and frequency table for this information.
Number of pieces
of homework
0
1
2
3
4
3
13
5
2
1
Remember that tally
marks are grouped
in 5’s.
Tally
Frequency
Exercise 3.4
A coach stops at a café.
Here is the passengers’ order:
Drink
a) How many adults ordered Orange juice?
Tea
//// //// /
b) How many Coffees were ordered?
Coffee
///
Adult
Orange ////
Juice
c) How many more adults than children
chose water?
Water
Child
////
//// //
//// //
//// //// ///
d) How many drinks were ordered
in total?
Mike asks pupils in his form group which lesson is their favourite.
science
P.E.
maths
English
science
P.E.
English
maths
English
science
P.E.
English
P.E.
science
maths
P.E.
English
P.E.
a) Draw a tally and frequency table for his information.
b) Which lesson is the most popular?
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Three pupils record in different ways the types of vehicle that pass their school.
Sue
Christine
bus car lorry
car bike
b
c
l
Matt
c
b
bus
car
lorry
bike
/
//
/
/
a) What problem may Sue have when she does her traffic survey?
b) What problem may Christine have when she does her traffic survey?
c) Why is Matt’s way the best to record this information?
This activity requires a coin.
Sam throws two coins 60 times recording the number
of tails shown for each throw.
Copy the table then carry out Sam’s experiment,
recording the results in your table.
Number of tails
0
1
‘I will get about 20
“no tails”, about 20 “1 tail”
and about 20
“2 tails’.
2
Tally
Frequency
Is Sam correct? Explain your answer.
Investigation
Requires children’s building blocks.
Ben thinks that if a building block is thrown, it will not land on
each face with the same frequency.
Carry out an experiment with different sizes of building block to
test his hypothesis.
Use a table with these headings to record your results:
Size of face
Small
Medium
Large
Use your results to decide if Ben is correct.
Use the results from your investigation in your last maths lesson.
Make a tally chart for each number in the two tables.
Adding the
numbers
Tally
Subtracting
the numbers
Tally
Frequency
Frequency
0
1
0
0
2
3
4
5
6
7
8
9
10 11 12
0
0
0
0
0
0
0
Total
frequency
Rhys says the number 4 has the highest probability of being a result.
Use your completed table to say whether Rhys is right or wrong.
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Key words
Estimating probability
estimate
Estimate probability from experiments.
When we carry out an experiment we can use the results to find the estimated
probability of each outcome. For example, if a coin is thrown 10 times and 6 tails are
recorded, the estimated probability of the coin landing on tails is 160 or 0.6.
Example 1
A sweet machine dispenses fruit lollipops.
The fruit is randomly selected by the machine.
Di records the frequency of each fruit.
Fruit
Orange
Cherry
Lemon
Strawberry
Frequency
2
4
8
6
a) What is the estimated probability of each fruit?
b) Which fruit is the machine most likely to select next?
a)
Fruit
Orange
Cherry
Lemon
Strawberry
Frequency
2
4
8
6
Estimated
probability
2
20
4
20
8
20
6
20
0.1
0.2
0.4
2 20 0.1
0.3
b) Lemon
Example 2
Ellen plays lucky dip at the fair. Each go costs 10p, and she has a chance of
winning 10p, winning 20p, or losing.
Frequency
Win 10p
Win 20p
Lose
5
1
4
Estimated probability
a) Copy and complete the table showing the estimated probabilities of
winning and losing.
b) How much did Ellen spend on the lucky dip?
c) How much did Ellen win?
d) How much in total did Ellen lose playing the game?
a)
Win 10p
Win 20p
Lose
Frequency
5
1
4
Estimated probability
5
10
1
10
4
10
21
b) Ellen spent 10 10p £1
c) Ellen won 5 10p 1 20p 50p 20p 70p
d) Ellen lost £1 70p 30p
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Exercise 3.5
Three dice are thrown.
Number of dice showing
the same outcome
3
2
0
Frequency
2
20
28
Estimated probability
a) How many times were the three dice thrown?
b) Copy and complete the table showing the estimated probabilities.
Ten counters are placed in a bag. Bindi chooses a counter at random, notes the colour
then puts the counter back. She repeats this 20 times.
Colour of counter
Blue
Red
Yellow
Frequency
11
6
3
Estimated probability
a) Copy and complete the table to find the estimated probability of choosing each colour.
b) How many of each colour of counter are likely
Remember that there are 10 counters.
to be in the bag?
There are two likely answers.
Dan plays a game at a school fair. Here is what he won and lost.
Outcome
Win 20p
Win 50p
Lose
Frequency
3
1
6
Use Example 2 to help.
Estimated probability
a)
b)
c)
d)
e)
How many times did Dan play the game?
Copy and complete the table for the estimated probabilities.
It costs 10p to play each time. How much did Dan spend?
How much money did Dan win?
Did Dan lose or win money overall? How much?
This question requires a dice.
Copy this table:
Number of throws
Tally
Frequency
Estimated probability
1
2
3
4
5
6
If you throw 2, 2, 5 it
has taken you 3
throws to total 6 or
more, so you would
put a tally mark in
the 3 column.
Throw the dice, adding up the numbers thrown until you reach a total of 6 or more.
Record in a table the number of throws needed to get this total. Repeat this experiment 50
times. Complete the table by finding the estimated probabilities for each number of
throws. Write each answer as a fraction and a decimal. For example, if it took you
3 throws only once out of the 50 tries, the estimated probability would be 510 or 0.02.
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Key words
Comparing probability
biased
theoretical
experimental
Compare theoretical and experimental probabilities
If you toss a fair coin, there is an equal chance of getting a head or a tail. But if the coin
was biased (or unfair), there would be a greater chance of either heads or tails. We
know that the probability of throwing a ‘2’ on a fair dice is 16. This is the theoretical
probability. When we actually throw the dice a number of times we can use our results
to calculate the experimental probability.
We do not expect the values for the theoretical and experimental probabilities to always
be the same.
Example
Caitlin throws two coins and counts the number of tails showing for each throw.
a) Find the theoretical probability for each outcome.
b) Caitlin throws the two coins 100 times. Here are her results:
Number of tails
0
1
2
Frequency
23
49
28
Find the experimental probability for each outcome.
c) Compare the theoretical and estimated probabilities, to find out if the
coin is fair.
a) Number of tails
0
1
2
Outcome
head, head
head, tail
tail, head
tail, tail
Theoretical probability
1
4
1
4
41 21
1
4
23
49
b) Experimental probability of 0 tails is 100, of 1 tail is 100 and of
There are four outcomes.
28
2 tails is 100.
c)
25
10 0
50
1
is the same as 41, and 100 is the same as 2, so the experimental
and theoretical probabilities are quite close. The coin is fair.
Exercise 3.6
Jane puts 20 counters in a bag: 10 blue, 6 red and 4 orange. She chooses a counter at
random, records its colour and returns it to the bag. She does this 20 times. Here are her
results:
Experimental probability
Blue
Red
Orange
9
20
6
20
5
20
a) What are the theoretical probabilities for each colour?
b) Jane says that she must have done the experiment wrong as the results for the
theoretical probabilities are not the same as the experimental ones. Is Jane correct?
Explain your answer.
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Activity for two players. Requires counters and a bag.
Pupil A chooses 10 counters of a mixture of blue and red colours and places them in a bag,
without Pupil B seeing them. Pupil B takes a counter from the bag, notes the colour and
replaces it. Repeat this 10 times, then estimate the number of each colour. Next, repeat the
experiment a total of 30 times, then 50 times. You can record your results in a table.
Number of counters taken from bag
10
30
50
Number of red counters chosen
Estimated probability
Number of blue counters chosen
Estimated probability
Empty the bag to see how many counters of each colour there are.
Calculate the theoretical probabilities then compare the theoretical probabilities with the
experimental probabilities for 50 repeats of the experiment. Write a sentence about your
results.
Requires square card and a cocktail stick.
Colour your spinner like the one shown but when you have made the
spinner, do not place the cocktail stick in the centre of your spinner,
so it is biased or not fair.
Spin your spinner 40 times, recording the colour it lands on.
Copy and complete the table. Calculate the probability of the
spinner landing on each colour if it was not biased.
Colour
Red
Blue
White
Yellow
Tally
Frequency
Estimated probability
Probability of landing on
colour if not biased
Requires random number tables or a random function on a calculator.
Charlie says that a number chosen at random has the same probability of being odd or even.
Odd
Even
Tally
Frequency
Experimental probability
Theoretical probability
0 is an even number
for this activity.
Use the random number table or the random function on a calculator to simulate 40 odd
or even numbers, by recording if the last digit is odd or even.
For example the number 0.3072 would be recorded as even as it ends with a 2.
a) How many numbers out of 40 would you expect to be odd if Charlie is correct?
b) Copy and complete the table showing each type of probability.
Comparing probability 37