© boardworks ltd 2006 1 of 55 d4 probability ks3 mathematics

© Boardworks Ltd 2006 1 of 55

D4 Probability

KS3 Mathematics


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D4.1 The language of probability

Contents

D4 Probability

D4.5 Experimental probability

D4.2 The probability scale

D4.4 Probability diagrams

D4.3 Calculating probability


The language of probability

Probability is a measurement of the chance or likelihood of an event happening.

Probability is a measurement of the chance or likelihood of an event happening.

Words that we might use to describe probabilities include:

unlikely50-50

chance likely

possible

probable

certain

poor chance

impossible

very likely

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:

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 abag a bag cbag cbag bbag bbag bbag b


Fair games

A game is fair if all the players have an equal chance of winning.

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).

No, this game is not fair. Team C is more likely to win.

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).


Fair games

A spinner has five equal sectors numbered 1 to 5.The spinner is spun many times.If the spinner stops on an evennumber team A gets 3 points.If the spinner stops on an odd number team B gets 2 points.

1

23

4

5

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 pointsTeam B would score 30 × 2 points = 60 points

Yes, this game is fair.


Scratch cards

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?

£ nowin

nowin

nowin

nowin

nowin

£ £ nowin

nowin £ no

win

£ nowin

nowin

nowin

nowin

nowin

nowin

£

nowin

nowin

nowin

nowin

nowin £ £

nowin

nowin £ no

win

nowin

nowin

£ nowin

nowin £

Scratch off a £ sign and win £10!

£ nowin

nowin

nowin

nowin

nowin

£ £ nowin


Bags of counters

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?

Choose a blue counter and win a prize!

bag abag a bag bbag b bag cbag cbag cbag c


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’ve just thrown four heads in a row so I’m much less likely to get a head on my next throw.

I’m so unlucky. If I roll this dice I’ll never get a six.

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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D4 Probability






The probability scale

The chance of an event happening can be shown on a probability scale.

impossible certaineven chanceunlikely likely

Less likely More likely

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



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:

Probabilities are written as fractions, decimal and, less often, as percentages between 0 and 1.

0 ½ 1impossible certaineven chance


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D4 Probability






Higher or lower


Listing possible outcomes

When you roll a fair dice you are equally likely to get one of six possible outcomes:

16

16

16

16

16

16

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?

P(tails) = 12

P(red) = 14

P(7 of ) = 152

P(Friday) = 17

2) This spinner stopping on the red section?

3) Drawing a seven of hearts from a pack of 52 cards?

4) A baby being born on a Friday?

1) A coin landing tails up?



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?

P(green) =310

or 0.3 or 30%



This spinner has 8 equal divisions:

a) a red sector?b) a blue sector?c) a green sector?

What is the probability of the spinner landing on

a) P(red) =28

=14

b) P(blue) =18

c) P(green) =48

=12



A fair die is thrown. What is the probability of gettinga) 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) = 16

b) P(a multiple of 3) = 26

=13

c) P(an odd number) = 36

=12



A fair die is thrown. What is the probability of gettinga) 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) = 36

e) P(a number bigger than 6) =

f) P(an integer) = 66

= 1

=12

0

Don’t write 0

6



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

Number of pupils

0

4

1

8

2

9

3

4

4

3

5

1

6

0

7

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) =930

=310



A bag contains 12 blue balls and some red balls.

The probability of drawing a blue ball at random from the

bag is .

How many red balls are there in the bag?

37

12 balls represent of the total.37

So, 4 balls represent of the total17

and, 28 balls represent of the total.77

The number of red balls = 28 – 12 = 16


The probability of an event not occurring

If the probability of an event occurring is p then the probability of it not occurring is 1 – p.If the probability of an event occurring is p then the probability of it not occurring is 1 – p.

The following spinner is spun once:

What is the probability of it landing on the yellow sector?

P(yellow) =14

What is the probability of it not landing on the yellow sector?

P(not yellow) =34



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 .

What is the probability of not pulling out a picture card?

3

13

P(not a picture card) = 1 – =313

1013



The following table shows the probabilities of 4 events. For each one work out the probability of the event not occurring.

EventProbability of the event occurring

Probability of the event not occurring

A

B

C

D

3

5

9

20

0.77

8%

2

5

11

20

0.23

92%



There are 60 sweets in a bag.

What is the probability that a sweet chosen at random from the bag is:

a) Not a cola bottle56

P(not a cola bottle) =

b) Not a teddy4560

P(not a teddy) =

10 are cola bottles, 14

are fried eggs,

the rest are teddies.20 are hearts,

=34


Mutually exclusive outcomes

Outcomes are mutually exclusive if they cannot happen at the same time.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

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 B: Tail

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.

Outcome C: the pupil has black hair.

Outcome D: the pupil has wears glasses.

These outcomes are mutually exclusive because a pupil can either have brown eyes, blue eyes or another colour of eyes.

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.

If two outcomes are mutually exclusive then their probabilities can be added together to find their combined probability.

What is the probability that a card is a moon or a sun?

P(moon) =13

and P(sun) =13

Drawing a moon and drawing a sun are mutually exclusive outcomes so:P(moon or sun) = P(moon) + P(sun) =

13

+13

= 23

For example, a game is played with the following cards:


Adding mutually exclusive outcomes

What is the probability that a card is yellow or a star?

P(yellow card) =13

and P(star) =13

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.

P(yellow card or star) =

We have to subtract the probability of getting a yellow star.

13

+13

–19

=3 + 3 – 1

9=

59


The sum of all mutually exclusive outcomes

The sum of all mutually exclusive outcomes is 1.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 = 0.9

P(red) = 1 – 0.9 = 0.1



A box contains bags of crisps. The probability of drawing out the following flavours at random are:

P(salt and vinegar) =25

P(ready salted) =13

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) =25

+13

=6 + 5

15=

1115

P(cheese and onion) = 1 –1115

=4

15



A box contains bags of crisps. The probability of drawing out the following flavours at random are:

P(salt and vinegar) =25

P(ready salted) =13

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 = 25

of 30 = 12 packets

Number of ready salted =13 of 30 = 10 packets

Number of cheese and onion =415 of 30 = 8 packets


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D4 Probability






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 equally likely outcomes.TT.HH, HT, TH,



2) We can use a two-way table.

Second coin

H T

H

TFirstcoin

HH HT

TH TT

From the table we see that there are four possible outcomes one of which is two heads so,

P(HH) =14 .



3) We can use a probability tree diagram.

First coinH

T

Second coinH

TH

T

Outcomes

HH

HTTH

TT

Again we see that there are four possible outcomes so,

P(HH) =14 .


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, 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

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.

8

8

8

8

8P(8) =

536


Scissors, paper, stone

In the game scissors, paper, stone two players have to show either scissors, paper, or stone using their hands as follows:

The rules of the game are that:

scissors paper stone

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.



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.

Scissors Paper Stone

Scissors

Paper

Stone

First player

Second player

SS SP ST

PS PP PT

TS TP TT

P(same hands) =

SS

PP

TT

39

=13



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.


Scissors

Paper

Stone

First player

Second player

SS SP ST

PS PP PT

TS TP TT

P(first player wins) =39

=13

SP

PT

TS



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.


Scissors

Paper

Stone

First player

Second player

SS SP ST

PS PP PT

TS TP TT

P(second player wins) =39

=13

ST

PS

TP



Is scissors, paper, stone a fair game?

P(first player wins) =13

P(second player wins) =13

P(a draw) =13

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?


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D4 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.

If we repeated the survey with a different sample the results would probably be slightly different.

From this we can estimate the probability of someone being

left-handed as or 0.087.87

1000

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.

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:

Relative frequency =Number of successful trials

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 =65100

=1320


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

Is the dice fair?

312002720038200302004220032200

= 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.

Expected frequency = theoretical probability × number of trialsExpected 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 .16

So, expected frequency = × 300 = 16

50

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

If you tossed a coin 250 times how many times would you expect to get a tail?

Expected frequency = × 250 = 12

125

If you rolled a fair dice 150 times how many times would you expect

to a number greater than 2?

Expected frequency = × 150 = 23

100


Spinners experiment


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 can be different.

The more times an experiment is repeated the more accurate the estimated probability will be.

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.

Jenny throws a dice 12 times and doesn’t get a six. She concludes that the dice must be biased.

© boardworks ltd 2006 1 of 55 d4 probability ks3 mathematics

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