3.1 Probability Key words - Pearson Education · Probability Find the probabilities of events Understand that if the probability of an event happening is p, the probability of it

26 Maths Connect 2B

An , such as rolling a dice, can have different .

When a fair six-sided dice is thrown, the possible outcomes are 1, 2, 3, 4, 5 and 6. Eachoutcome is equally likely. The probability of throwing a 2 is �

16�, as there is a 1 in 6 chance

of this happening.

It is certain that one of the numbers 1 to 6 will be thrown. If an outcome is certain, ithas a probability of 1. So, if the probability of throwing a 2 is �

16�, the probability of

throwing a number other than 2 is 1 � �16�. We write this as:

P(2) � �16�

P(not 2) � 1 � �16� � �

56�

outcomesevent

ProbabilityFind the probabilities of events

Understand that if the probability of an event happening is p, the probability of it not

happening is 1 � p

Key wordseventoutcome

3.1

Example 1 A bag contains discs numbered from 1 to 10. One disc is chosen at random.What is the probability that the number is: a) even b) not even c) less than 7d) not less than 7 e) a prime number f) not a prime number?

Show your answers as a fraction, decimal or percentage.

a) �150� � 0.5

c) �160� � �

35�

e) �140� � 40%

The prime numbersbetween 1 and 10 are 2, 3,5 and 7.

Example 2 The probability that Pran is late for school in the morning is 0.1.a) What is the probability that he is not late for school?b) How many times would Pran expect to be late in 20 mornings?

a) 1 � 0.1 � 0.9

b) 0.1 � 20 � 2. Pran would expect to be late twice in 20 mornings.

Exercise 3.1 ..........................................................................................� 50 raffle tickets are sold, numbered 1–50. A ticket is chosen at random. What is the

probability that the number is:a) even b) less than 10c) not less than 10 d) odd and divisible by 5e) odd and divisible by 2 f) not number 6?Show your anwers as percentages.

b) 1 � �21

� � �21

� � 0.5

d) 1 � �160� � �

1100� � �1

60� � �1

40�

f) 1 � �140� � �

1100� � �1

40� � �1

60� � 60%

03 Section 3 pp026-037.qxd 9/1/04 9:02 am Page 26

� A fair dice is thrown. Copy and complete the table to show the probabilities of eachevent. Give your answers as fractions.

� A CD contains 15 tracks. The tracks are played at random. What is the probability thatthe first track is a) number 6 b) not number 6?

� Copy and colour the spinner so that the probability of it landing on:a) green is �

13�, not blue is �

56�, not white is �

12�

b) green is �13�, not blue is �

56�, not white is �

23�.

� The weather forecast predicts that tomorrow there will be a 65% chance of rain. What isthe probability that it will not rain?

� There are 20 horses in a field coloured grey, chestnut, bay or black. If a horse is chosen atrandom, the probability of bay is 0.5, of grey is 0.25 and of not black is 0.9. How many ofeach colour horse are there?

� A bag contains 8 sweets. The probability of a caramel being pickedis 0.5. A caramel is removed from the bag and not replaced. A sweetis chosen at random. What is the probability that it is a caramel?

� A school sells 100 raffle tickets numbered 1–100. The winning ticket is drawn at random.a) Mair has ticket numbers 15, 16, 17, 18, 19. Sean has tickets number 25, 35, 45, 55, 65.

Who has the better chance of winning? Explain your answer.b) Russell has some tickets. He has a 15% chance of winning. How many tickets does he

have?

The probability of an archer hitting a target with an arrow is 0.9.Thirty arrows are fired. How many arrows are expected to hit thetarget?

27Probability

Event Probability of event Probability of event NOT happening happening

Throwing a 6

Throwing an odd number

Throwing a number less than 3

Throwing a prime number

Throwing a number more than 6

Using counters maybe helpful.

Kim and Selim have four cards each. Kim has numbers 1, 2, 3, 4 and Selim has 5, 6, 7, 8. They each remove one card, then add together the numbers on theirremaining three cards. Investigate these probabilities for Kim and Selim:a) the total is an odd number b) the total is greater than 7.

Investigation

Look at Example 2.

You need four colours.


28 Maths Connect 2B

We can show all the possible outcomes of an event on a diagram.

If several outcomes are all equally likely, we say they are .random

sample space

Diagrams and tablesTo record all possible outcomes of events systematically

Key wordssample spacerandom

3.2

Example 1 A coin can show a head (H) or a tail (T). Two coins are thrown. What are thepossible outcomes?

Coin 1 Coin 2

H H

H T

T H

T T

Example 2 Two tetrahedral dice numbered from 1 to 4 are thrown. The total of thenumbers lying face downwards is found and recorded.a) List all the possible outcomes.b) What is the probability that the total of the two dice is:

i) 5 ii) even iii) less than 4 iv) not less than 4 v) 1.

The sample space is theshaded numbers.

� 1 2 3 4

1 2 3 4 5

2 3 4 5 6

3 4 5 6 7

4 5 6 7 8

a) The possible outcomes are 2, 3, 4, 5, 6, 7 and 8.

b) i) �146� � �4

1� ii) �1

86� � �2

1� iii) �1

36� iv) 1 � �1

36� � �1

163� v) 0

Exercise 3.2 ..........................................................................................� a) A teacher chooses two pupils from her class. Using B for a boy and G for a girl,

show the four possible outcomes using a sample space diagram.b) Find the possible outcomes if the teacher chooses three pupils.

� Copy and complete the table to show all the possible outcomes when a dice is thrownand a coin is spun.

Dice1 2 3 4 5 6Coin

heads (H) 1, H

tails (T)


� Luke has digit cards 1, 3 and 5. Adam has digitcards 2, 4 and 6. They each choose a card atrandom and add their two numbers together.Copy and complete the sample space diagram toshow all of the possible outcomes.Use it to find the probability that the total will be:a) 5 b) less than 6 c) not less than 6 d) even.Use a sample space diagram to show all of the possible outcomes if the numbers aremultiplied together.

� Matt has the digit cards 4, 5 and 6.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 ii) not less than 50 iv) divisible by 5?

� Maggie has digit cards 1, 3, 4 and 9.a) How many possible 2-digit numbers can she make?b) What is the probability that a 2-digit number chosen at random is:

i) odd ii) less than 50 ii) not less than 50 iv) divisible by 3?

� Sangheeta and George throw three coins. Sangheeta wins if two or more heads show.George wins if less than two heads show. Is this a fair game? Explain your answer.

� a) Copy and complete the sample space diagram to show all possible outcomes when these two spinners are spun. The spinner that lands on the highest number wins. Use ‘A’ to show that A wins, ‘B’ to show B wins and ‘�’ to show that both numbers were the same.

Does each spinner have the same chance of winning? Explain your answer.

b) Repeat part a) using these twospinners.

45

5

2 2

3

Spinner R

45

5

2 4

4

Spinner S

45

6

1 2

3

Spinner A

35

5

1 1

3

Spinner B

29Diagrams and tables

� 1 3 5

2 3

4

6

� Use the idea from Q7 to design two different spinners so that the probability of eitherspinner winning is equal.

Investigation

Spinner A1 2 3 4 5 6Spinner B

1 �

1 A

3

3

5 B

5


30 Maths Connect 2B

In Lesson 3.2, we saw that the possible outcomes of throwing two coins are:

head, head

head, tail

tail, head

tail, tail

We can also use a diagram called a to show all posssible outcomes oftwo or more events.

head

1st coin 2nd coin

tail

head

tail

head

tail

outcome

head, head

head, tail

tail, head

tail, tail

tree diagram

Tree diagramsDraw tree diagrams to show all posssible outcomes

Key wordstree diagram3.3

Example An ordinary dice is thrown when a coin is spun. Draw a tree diagram toshow all of the possible outcomes.

dice coin

1

2

3

4

5

6

head

tail

head

tail

head

tail

head

tail

head

tail

head

tail

outcome

1, head

1, tail

2, head

2, tail

3, head

3, tail

4, head

4, tail

5, head

5, tail

6, head

6, tail


Exercise 3.3 .............................................................................................� Copy and complete the tree diagram

to show all possible outcomes when these two spinners are spun.

� A football match can finish in a ‘win’, ‘loss’ or ‘draw’. Draw a tree diagram to show allposssible outcomes when two football matches are played.

� Yasmin wants to go to two clubs during the week. On Monday she can choose swimming (S), kick boxing (K), guitar (G) or tai chi (T). On Wednesday she can choosechess (C) or rock climbing (R). Draw a tree diagram to show all the possible combinationsof clubs.

� A restaurant has a choice of potatoes and vegetables with lunch. Draw a tree diagram to show possible choices of one type of potato and one vegetable.

� Bag A contains a red and a blue counter. Bag B contains a green, a black, a white and ayellow counter. Draw a tree diagram to show all the possible outcomes when a counter istaken at random from each bag.

� A bag contains three cubes: a blue, a yellow and a white. A cube is removed from the bag,its colour recorded and then it is replaced. Another cube is removed from the bag. Draw atree diagram to show all possible outcomes of the colours of the two cubes removed fromthe bag.

Spinner 1 Spinner 2

green

yellow

black

Outcome

red

blue

white

green, red

green, blue

green, white

Spinner 1 Spinner 2

31Tree diagrams

� Look at the tree diagrams you have drawn. Count the total number of outcomes foreach one. What is the connection between the number of possible outcomes for eachindividual event and the total number of outcomes for both events? Use your answerto predict the total number of outcomes when a coin is spun and a 4-sided dice,numbered 1–4 is thrown. Draw the tree diagram to check if your prediction is correct.

Investigation

Spinner 1 Spinner 2


32 Maths Connect 2B

For a fair, six-sided dice, the of throwing a six is �16�. We find

the theoretical probability by using the possible outcomes.For a particular six-sided dice, we could find the byexperimenting. This might be different from the theoretical probability.

The estimated probability must be between 0 and 1 inclusive and written as a fraction,decimal or percentage. If a coin is thrown 50 times and 28 tails are recorded, theestimated probability of the coin landing on tails is �

2580�, 0.56 or 56%.

estimated probability

theoretical probability

Estimating probabilityTo estimate probability from experiments

Key wordstheoretical probabilityestimated probability

3.4

Example Three coins are thrown repeatedly and the number of heads each time isrecorded.

a) How many times were the coins thrown?b) Find the estimated probability for each number of heads thrown.c) Use the answers to part b) to estimate the probability

of three coins showing two or more heads.

Number of heads 0 1 2 3

Frequency 12 14 15 9

a) 50 times

b)

c) 0.3 � 0.18 � 0.48

Number of heads 0 1 2 3

Estimated probability �5120� � 0.24 �5

140� � 0.28 �5

150� � 0.3 �5

90� � 0.18

Exercise 3.4 ..........................................................................................� A dice is thrown and the number showing is recorded.

a) How many times was the dice thrown?b) Find the estimated probability for each of the six numbers.c) What is the estimated probability that the dice will show:

i) An even number ii) A number less than 4 iii) Not a 2?

� Rick shuffles a pack of postcards, turns the top card over and records the type ofpicture in a frequency table.

Find the estimated probability for each type of card. Show your answers as decimals.

Number 1 2 3 4 5 6

Frequency 12 8 7 9 14 10

Type of card View Animal picture Transport picture

Frequency 5 21 14

12 � 14 � 15 � 9 � 50


� A class count the number of paper clips in 30 small boxes. They find that 26 boxescontain 51 or less paper clips. What is the estimated probability that the next box countedcontains more than 51 paper clips?

� To win a prize at tombola, Jane has to pick from a drum a ticket ending in 0 or 5.Assuming that all ticket numbers from 1–500 are put in the drum at the start, what is theestimated probability of Jane winning a prize?

� Work with a partner. You will need dominoes.Share the dominoes equally and then each select a domino at random. Is there a ‘snap’ between any of the numbers? (Do not count a ‘double’ as a snap.)Repeat this experiment 50 times, then record your results in a table like this.

� This activity requires three dice.a) What is the lowest possible total when three dice are thrown together? The highest total?b) Use your answers to part a) to complete a tally chart showing

all of the possible totals when three dice are thrown.c) Throw the three dice a total of 50 times. Each time record the

total in your table.d) Find the estimated probabilities for each of the possible totals.

� This activity requires a dice.Throw a dice, adding up the numbers thrown until you reach a total of six or more.Record the number of throws needed to get this total in the table. You do not need torecord your total. Repeat this experiment 40 times.

Complete the table by finding the estimated probabilities for each amount of throws.

� This activity requires the spinner from worksheet 5.4, and a cocktail stick.Use the cocktail stick to make a hole in the spinner. Do not put the hole in the centre ofthe spinner. (The idea is to make the spinner biased.) Spin your spinner 60 times,recording the number it lands on. Copy this table and use it to record your results.

Comment on your results.

33Estimating probability

Snap!

Snap No snap

Tally

Frequency

Estimated probability

Total

Tally

Frequency


Number of throws for 1 2 3 4 5 6a total of 6 or more

Tally

Frequency


Number 1 2 3 4 5 6

Tally

Frequency



34 Maths Connect 2B

A dice is thrown six times with these results:

The number thrown is or .We would not say that is impossible to throw a 6 based only on these throws as wehave not thrown the dice enough times.When doing an experiment, in general, the more times we perform the experiment themore our results will be.Also, if we throw the dice another 6 times we do not expect to get exactly the sameresults each time.

reliable

unpredictablerandom

Better estimates of probabilityIf an experiment is repeated there will usually be different results

Increasing the number of times an experiment is performed will usually

give a better estimate of probability

Key wordsrandomunpredictablereliable

3.5

Example Helen throws a coin 10 times and counts the number of heads and tails.

She throws the coin a further 50 times with these results:

a) Calculate the estimated probabilities for both frequency tables.b) Draw a bar-line graph showing these probabilities.

Number of tails 6

Number of heads 4

Number of tails 23

Number of heads 27

a)

b)


Number of tails 6 0.6

Number of heads 4 0.4


Number of tails 23 0.46

Number of heads 27 0.54

Estimated probabilities of acoin showing heads or tails

Est

imat

edp

rob

abil

ity

No. of t

ails a

fter

No. of t

ails a

fter

No. of h

eads a

fter

No. of h

eads a

fter

10 th

rows

50 th

rows

10 th

rows

50 th

rows

00.10.20.30.40.50.60.7


Exercise 3.5 .............................................................................................� This activity requires two dice.

Throw the two dice 10 times, recording the total for each throw in the table below.

Draw a bar-line graph for your results, plotting the total on the x-axis and the frequency on the y-axis. Repeat this experimentfor a total of 20 throws, and then for a total of 50 throws. Eachtime draw a bar-line graph.Compare the shape of your three graphs and write a sentenceabout your results.

� This activity requires a coin.Look at the Example. Do Helen’s experiment for yourself. Record your results for 10, 30and 50 throws of the coin. Draw a bar-line graph to show your results.

� This activity requires ‘Lego’ bricks.What is the probability of a piece of ‘Lego’landing ‘face up’, ‘face down’ or ‘on one of itssides’?Decide what information you are going to record and draw a table. Throw the ‘Lego’ 10times, 30 times and 50 times. For each set of throws, find the estimated probability of itlanding on a particular side.Repeat this activity using a different sized piece of ‘Lego’.Compare your results. Do different pieces of ‘Lego’ have different probabilities forlanding in a particular way?

� This activity requires a calculator with a ‘random’ button.To simulate throwing an ordinary dice:

● Use the ‘random’ button to get a decimal number between 0 and 1.● Multiply this by 6, then add 1.● Use the number before the decimal point to simulate the number on the dice.

For example: 0.632 � 6 � 1 � 4.792. This is the same as throwing a 4 on a dice.

Use this method to generate 60 ‘throws’ of a dice, recording your results in a table.

Compare your results to the theoretical frequencies of throwing numbers 1–6 on anordinary dice.

35Better estimates of probability

Total 2 3 4 5 6 7 8 9 10 11 12

Tally

Frequency

You can use your first 10throws as part of your 20,and your 20 as part ofyour 50.

Random number 1 2 3 4 5 6

Tally

Frequency

Theoretical frequency �16� �

16� �

16� �

16� �

16� �

16�


36 Maths Connect 2B

Throwing a dice produces random outcomes. We can find the

of throwing a 6 and also, by throwing the dice, find the . We do

not always expect these two values to be the same.

estimated probability

theoretical probability

Comparing probabilitiesCompare theoretical with experimental probabilities

Key wordstheoretical probabilityestimated probability

3.6

Example Three coins are thrown and the number of tails counted.Copy and complete the table to show:a) the possible outcomes b) the theoretical probability for each outcome.

Caitlan throws three coins 100 times. Here are her results:

c) Estimate the probability for each outcome.d) Compare the theoretical and estimated probabilities.

Number of tails 0 1 2 3

a) Outcome

b) Theoretical probability


Frequency 12 39 35 14

d) The theoretical and estimated probabilities were quite close, but we would not expect

them to be exactly the same. Increasing the number of times the experiment was

performed may improve the accuracy of the estimated probabilities.


a) Outcome HHH HHT TTH TTTHTH THTTHH HTT

b) Theoretical probability �81� � 0.125 �8

3� � 0.375 �8

3� � 0.375 �8

1� � 0.125

c) Estimated probability �1012

0� � 0.12 �13090� � 0.39 �1

3050� � 0.35 �10

140� � 0.14

Exercise 3.6 ..........................................................................................� Four pupils each throw two dice. They record the number

of times the two numbers are the same.

Name Number of throws Both dice same Both dice different

Falek 30 4 26

Paolo 120 17 103

Alan 70 13 57

Nai 180 22 158


a) Which pupil’s results are most likely to give the best estimates of the probabilities?Explain your answer.

b) Complete the table to show the results of all 400 throws.

c) Draw a table to show all the possible outcomes when two dice arethrown. Use ‘S’ to show that the dice are the same, ‘D’ to show theyare different.

d) What is the theoretical probability that the two dice:i) show the same number? ii) show different numbers?

e) Use the theoretical probabilities to calculate for 400 throws the number of times thedice will be:iii) the same iv) different.

f) Compare the pupils’ results with the theoretical results. Why are they not the same?

� a) This activity is for two and requires dice andcounters. Each player takes turns to throw twodice, adding the scores together. If the total is oneof their four categories of outcomes, a counter isplaced on that square. The first player to cover allfour of their squares wins.

b) Draw a sample space diagram to show all the possible outcomes when two dice are thrown and the scores added together.Use the theoretical probabilities to show that the game in part a) is not a fair game.

c) Design a similar game where both players have an equal chance of winning.

� This activity is for two players and requires counters and a bag.One pupil chooses 20 counters of a mixture of blue and red colours and places them in abag. The other pupil takes a counter from the bag, notes the colour and replaces it.Repeat 10 times. Estimate the number of each colour counter. Repeat the experiment atotal of 30 times, then 50 times. Record your results in a table.

Empty the bag to see how close your estimates were.

37Comparing probabilities

Both dice same Both dice different

Number of throws

Estimate of probability

� 1 2

1 S D

2

Number of counters taken from the bag 10 30 50

Number of red counters


Number of blue counters


Even

Player A

Singledigit

Squarenumber

Multipleof 5

Primenumber

Player B

Odd

Factorof 8

Trianglenumber


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3.1 Probability Key words - Pearson Education · Probability Find the probabilities of events Understand that if the probability of an event happening is p, the probability of it