2

Understand the following

0 ( ) 1P x

number of favorable outcomesProbability =

total possible outomes

( ) 1P x

Consider each of the following statements

Declare how you would best define the probability that each will happen using one of the following words

UnlikelyImpossibleLikelyCertain Even Chance

The probability scale

0

Impossible

½ 1

CertainLikelyUnlikely Even chance

a) an event with a probability of 0.8 would be described as very likely

b) an event with a probability of 0.4 would be described as unlikely

c) an event with a probability of 1/20 would be described as very unlikely

d) an event with a probability of 6/12 would be described as even chance

Probability is a numerical measure of how likely or unlikely an event is to occur.

Probabilities are usually written as fractions, but can be written in any form equivalent to that fraction. Eg ¾ = 0.75 = 75%

Probabilities can be anywhere between 0 (impossible) and 1 (certain):

abc d

The probability scale

0 ½ 1

1. Complete this probability scale using the key words given

Key words:Even chanceLikelyImpossibleCertainUnlikely

2. Label the events described below on the probability scale:a) The chance of getting an even number when rolling a diceb) The chance of winning the National Lotteryc) The chance of rain in March

3.Describe an event that you think has a probability of:

a) 0.3 _________________________________________________

b) 1 _________________________________________________

c) 0.8 _________________________________________________

0 ½ 1

Impossible CertainLikelyUnlikelyEven

chance

ab c

Probability of an event

1. Bob is picking randomly from a bag containing tiles numbered 1 to 10.Write down the probability that the number he picks is:

a) 7 b) 4 or less c) Odd d) A multiple of 3

2. A survey is conducted of pupils’ favourite team:

Team Spurs Man Utd Liverpool Arsenal

Pupils 12 8 4 6

John picks a pupil at random to ask more questions.Write down the probability that the pupil he picks supports:

a) Liverpool b) A London team c) Not Liverpool

3. A bag contains 20 coloured balls, some red and some blue. Keith knows that the probability of picking a red ball is 2/5. How many red balls are there?

101

104

52 10

52

1 103

304

152 30

185

3 1513

5

2 4

4

208

8 red balls

3. The table shows information about the number of goals scored by Aston Villa in each game of the season so far:

Number of goals

Number of games

0 3

1 6

2 5

3 4

More than 3

2

Mr Walker has all the games on DVD and decides to watch one. He picks a game randomly. What is the probability he picks a game with:

a) exactly 1 goal by Aston Villa,20

6

10

3

20

11

4. The cumulative frequency curve below shows the distribution of the height of 50 students. Estimate the probability that a student picked at random will be more than 164cm tall

50

10

5

1

Total = 20 games

b) 2 or more goals by Aston Villa

Total probability of events

If one of a set of possible outcomes xi must happen, then ΣP(xi) = 1

The sum of their probabilities is 1

Eg the probability of rain today is 0.7 so the probability of no rain is 0.3

Eg a coin is biased so that the probability of heads is 3/5 so the probability of tails is 2/5

Eg a 4-sided spinner has the following probabilities of getting each number. The probability that the spinner will land on 2 is equal to the probability it will land on 4. Complete the table.

Number 1 2 3 4

Probability 0.2 0.3x x13020 xx ..

1502 .x

502 . x

250. x

3. The weatherman claims that it is twice as likely to snow as not.Complete the table:

Snow No snow

Probability 31

32

Total probability of events

2. A die is biased so that the probability of each number is:

Number 1 2 3 4 5 6

Probability 0.1 0.15 x 0.2 x 0.05

Find the value of x

1250 x.

250. xΣP(xi) = 1

1. A die is biased so that the probability of rolling a six is .What is the probability of not rolling a six?

41

43

411

At the races

2

3

4

5

6

7

8

9

10

11

12

Each horse moves 1 square if you get its total when you roll both dice. Is it a fair race?

Why isn’t the race fair?

1 2 3 4 5 6

1

2

3

4

5

6

2 3 4 5 6 7

3 4 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

5

Dice A

Dic

e B

outcomes possible of Number

outcomes desired of NumberyProbabilit

Consider the possible outcomes from finding the total of 2 dice:

Total Probability

361

362

363

364

365

366

365

361

362

363

364

7 is the most likely total, so horse 7 is most likely to win

2 & 12 are the least likely totals, so horses 2 and 12 are least likely to win

2

3

4

5

6

7

8

9

10

11

12

Probability using tables

1 2 3 4 5 6

1

2

3

4

5

6

1 2 3 4 5 6

2 4 8 10 12

3 6 9 12 15 18

4 8 12 16 20 24

5 10 15 20 25 30

6 12 18 24 30 36

6

Dice A

Dic

e B

Eg in a game, two fair dice are rolled and a score is found by multiplying the numbers obtained together.

a) Show the possible outcomes in the table belowb) Use your completed table to find the probability of getting a score of 12c) Use the table to find the probability of getting a score of 23 or more

a)b)

36

4yProbabilit

9

1

4 outcomes out of 36 give a score of 12

c)

36

6yProbabilit

6

1

6 outcomes out of 36 give a score of 23+

outcomes possible of Number

outcomes desired of NumberyProbabilit

Probability using tables

1 2 3 4

1

2

3

4

2 3 4 5

3 4 6

4 5 6 7

5 6 7 8

5

Spinner A

Sp

inn

er B

1. In a game, two fair spinners are spun and a score is found by adding the numbers obtained together.

a) Show the possible outcomes in the table belowb) Use your table to find the probability of getting a score of 7c) Find the probability of getting a score of 4 or less

a)b)

16

2yProbabilit

8

1

2 outcomes out of 16 give a score of 7

c)

16

6yProbabilit

8

3

6 outcomes out of 16 give a score of 4 or less

outcomes possible of Number

outcomes desired of NumberyProbabilit

R B Y

R

B

Y

RR BR YR

RB BB

RY BY YY

YB

1st pick

2nd p

ick

2. Keith has 3 coloured balls in a bag- red, blue and yellow.He picks one, records its colour, puts it back and picks another.a) Complete the table to show the possible outcomesb) Write down the probability Keith picks: i) two balls of the same colourii) two balls of different colouriii) at least one yellow ball

a)

bi)

9

3yProbabilit

3

1

3 outcomes out of 9 give the same colour

ii)

3

11yProbabilit

3

2

Either they are the same colour or not

outcomes possible of Number

outcomes desired of NumberyProbabilit

iii)

9

5yProbabilit

5 outcomes out of 9 have a yellow ball

Tree diagrams

Sometimes, a tree diagram can help you understand probabilities

Eg a coin is biased so that the probability of throwing heads each time is 2/3

H

H

H

T

T

T

32

32

32

31

31

31

1st throw 2nd throw

The branches show the possible outcomes and their probabilities

Any chain of branches from the beginning to the end represents a combination of outcomes

Two heads in a row

Heads followed by tails

Tails followed by heads

Two tails in a row

Tree diagramsEg Johnny has a 0.4 chance of scoring from a free-kick and a 0.7 chance of

scoring from a penalty

Score

Score

Score

Miss

Miss

Miss

40.

70.

70.60.

30.

30.

Free-kick Penalty

The branches show the possible outcomes and their probabilities

Any chain of branches from the beginning to the end represents a combination of outcomes

Scores both

Scores free-kick but misses penalty

Misses free-kick but scores penalty

Misses both

Finding probabilities with tree diagramsEg a coin is biased so that the probability of throwing heads each time is 2/3

H

H

H

T

T

T

32

32

32

31

31

31

1st throw 2nd throw

P(Two heads in a row)

P(Heads followed by tails)

P(Tails followed by heads)

P(Two tails in a row)

P(one head, one tail)

To find the probability of a combination of outcomes, multiply the probabilities along the relevant branches

32

32 9

4

31

32 9

2

32

31 9

2

31

31 9

1

If more than one combination gives the desired outcome, add their probabilities 9

29

2 94

Finding probabilities with tree diagramsEg Johnny has a 0.4 chance of scoring from a free-kick and a 0.7 chance of

scoring from a penalty

Score

Score

Score

Miss

Miss

Miss

40.

70.

70.60.

30.

30.

Freekick Penalty

P(scores both)

P(scores neither)

P(scores one)

To find the probability of a combination of outcomes, multiply the probabilities along the relevant branches

7040 .. 280.

3060 .. 180.

If more than one combination gives the desired outcome, add their probabilities

3040 ..

540.

7060 ..

Tree diagrams

4

33

1

S im o nw in s

S im o nd o es n o tw in

34

..........

13

..........

S im o nw in s

S im o nd o es n o tw in

..........

S im o nw in s

S im o nd o es n o tw in

..........

ten n is sn o o k er

1. Simon plays one game of tennis and one game of snooker.

The probability that Simon will win at snooker is

a) Complete the tree diagram

b) Work out the probability that Simon wins both games.

The probability that Simon will win at tennis is

c) Work out the probability that Simon will win only one game.

4

1

3

2

3

1

3

2

3

1

4

3

4

1

3

1

4

1

3

2

4

3

12

1

12

6

12

7

2. Julie and Pat are going to the cinema.The probability that Julie will arrive late is 0.2The probability that Pat will arrive late is 0.6The two events are independent.

a) Complete the diagram.Ju lie

P a t

la te

la te

la te

n o tla te

n o tla te

n o tla te

0 .2

0 .6

b) Work out the probability that Julie and Pat will both arrive late.

c) Work out the probability that neither of them arrive late.

80.

40.

60.

40.

6020 .. 120.

4080 .. 320.

3. Julie throws a fair red dice once and a fair blue dice once.

a) Complete the probability tree diagram to show the outcomes.

Label clearly the branches of the probability tree diagram.The probability tree diagram has been started in the space below.

16

R edD ice

B lu eD ice

S ix

N o tS ix

b) Calculate the probability that Julie gets at least one six.

6

5

6

5

6

51

36

251

36

11

Six

Six

Not

Not

Six

Six

6

5

6

5

6

1

6

1

= 1 – P(no sixes)

4. Loren has two bags.The first bag contains 3 red counters and 2 blue counters.The second bag contains 2 red counters and 5 blue counters.Loren takes one counter at random from each bag.

a) Complete the probability tree diagram.

35

R ed

B lu e......

R ed

B lu e......

27

R ed

B lu e......

......

C o u n ter fro mfirst b a g

C o u n ter fro mseco n d b a g

b) Work out the probability that Loren takes one counter of each colour.

5

2

7

5

7

2

7

5

7

2

5

2

7

5

5

3

35

4

35

15

35

19

1 2 3 4 5

6 7 8 9 10

Play your cards right

6 2 4 1 7 8 9

Play your cards right

9

4)(HigherP8

7

7

5

6

6

5

3

4

2

1 2 3 4 5 6 7 8 9 10

Higher or lower???The probability of higher or lower is conditional

on the cards that have already appeared

Non-replacementEg a bag contains 3 red and 7 blue balls.A ball is picked, not replaced, and another picked.Complete the tree diagram

If the object is not replaced, this affects the probabilities on the 2nd pick

Red

Red

Red

Blue

Blue

Blue

103

92

93

107

97

96

1st pick 2nd pick

If a red ball was picked first, there are only 2 red balls left

If a red ball was picked first, there are still 7 blue balls left

If a blue ball was picked first, there are still 3 red balls left

If a blue ball was picked first, there are only 6 blue balls left

There are 10 balls to choose from when picking the 1st ball

Non-replacementEg a bag contains 3 red and 7 blue balls.A ball is picked, not replaced, and another picked.Find the probability that:a) 2 red balls are pickedb) 1 of each colour is picked

To find the probability of a combination of outcomes, multiply the probabilities along the relevant branches

Red

Red

Red

Blue

Blue

Blue

103

92

93

107

97

96

1st pick 2nd pickP(both red)

P(one of each)

92

103 90

6

If more than one combination gives the desired outcome, add their probabilities

97

103

9042

151

9021

9021

93

107

157

Non-replacement1. A bag of sweets contains 2 toffees and 5 chocolates.A sweet is picked, eaten, and another picked.a) Complete the tree diagramb) Find the probability that:i) Both sweets are toffeesii) 1 of each sweet is picked

Toffee

Toffee

Toffee

Chocolate

Chocolate

Chocolate

72

61

62

75

65

64

1st pick2nd pick bi) P(both toffee)

bii) P(one of each)

61

72 42

2

65

72

2110

211

4210

4210

62

75

2. 5 white socks and 3 black socks are in a drawer.

Stefan takes out two socks at random.Work out the probability that Stefan takes out two socks of the same colour.

White

White

White

Black

Black

Black

85

74

75

83

73

72

P(both white) 74

85 56

20

P(both black) 72

83 56

6

P(same colour) 566

5620

5626

2813

1st pick 2nd pick

Eg A bag contains 3 blue beads, 5 yellow beads and 2 green beads.Sid takes a bead at random from the bag, records its colour and replaces it.He does this two more times.

Work out the probability that, of the three beads Sid takes, exactly two are blue.

Listing outcomes systematically

BBY

BYB

YBB

BBG

BGB

GBB

and P(BBY) = P(BYB) = P(YBB)

and P(BBG) = P(BGB) = P(GBB)

10

5

10

3

10

3P(BBY)

1000

45

10

2

10

3

10

3P(BBG)

1000

18

So P(exactly 2 blue) 31000

183

1000

45

1000

189

Combinations with exactly 2 blue:

A tree diagram would take too long here

Listing outcomes systematically

1. For any match, the probabilities of each result for Aston Villa are as follows:

P(win) = 2

1P(draw) =

3

1P(lose) =

6

1

Find the probability that, in 3 matches, Aston Villa win exactly 2 matches

WWD

WDW

DWW

WWL

WLW

LWW

Combinations with exactly 2 wins:

and P(WWD) = P(WDW) = P(DWW)

and P(WWL) = P(WLW) = P(LWW)

3

1

2

1

2

1P(WWD)

12

1

6

1

2

1

2

1P(WWL)

24

1

So P(exactly 2 wins) 324

13

12

1

8

3

2. A bag contains 2 blue balls and 3 green balls.Pete takes a ball at random from the bag, records its colour and replaces it.He does this two more times.

Work out the probability that, of the three balls Pete takes, exactly two are the same colour. (Hint – what is the alternative to 2 being the same colour?)

‘2 the same colour’ means 1 is a different colour

The only other option is all 3 are the same colour

So P(2 the same colour) = 1 – P(all the same colour)

P(all the same colour) = P(BBB) + P(GGG)3

5

2

3

5

3

125

27

125

8

125

35

25

7

So P(2 the same colour) = 25

71

25

18

Expectation

Expectation is the long-run average you would get if a test was repeated many times

If an event has probability p, the expectation in n trials is np

Expectation is used as an estimate for how many times an event will occur

Eg a coin is biased so that the probability of throwing heads is ¾. Dave is going to throw the coin 200 times. Work out an estimate for the number of times the coin lands on heads.

so expectation = np = 200 x ¾ = 150n = 200 and p = ¾

Eg There are 306 MPs in the Conservative Party. 4/5 of them say they support proposals to increase tuition fees.Work out an estimate for the number who will vote in favour of the changes

so expectation = np = 306 x 4/5 = 244.8 = 245 to nearest integern = 306 and p = 4/5

Expectation1. A coin is biased so that the probability of getting heads is 3/5. Dave is going to throw the coin 120 times. Work out an estimate for the number of times the coin lands on tails.

Expectation = np = 120 x 2/5 = 48

2. The chance of rain each day in April is 2/3. Estimate the number of days you can expect rain in April.

3. A door-to-door salesman achieves sales with a probability of 3/10.How many doors must he approach in order to expect an average of 15 sales a day?

Expectation = np = 30 x 2/3 = 20

If 3/10 x n = 15

30 days in April

then n = 10315 = 50 doors

Keith designs a game. It costs £1.60 to play the game.

The probability of winning the game is 2/5

The prize for each win is £3

80 people play the game.

Work out an estimate of the profit that Keith should expect to make.

Expectated winnings

Expected winners = 2/5 x 80 = 32

Expected payout = 32 x 3 = £96

Takings = 80 x 1.6 = £128

Estimated profit = 128 - 96 = £32

Profit = takings - costs

Expectation = np

John

JohnJohn

Tom

TomTom

52

43

43

53

41

41

5. John and Tom play darts and pool every Saturday. John wins at darts 2/5 of the time and wins at pool ¾ of the time.a) Find the probability they win one of the games each.Estimate the number of times they win one of the games each, over a 60 week period

Winner at darts

Winner at pool a)

4

3

5

3

4

1

5

2

20

9

20

2

20

11

Expectation = np

20

1160

= 33 times

b)

4. A fruit machine costs £1 to play and pays out £40 with a probability of 1/20.Is the machine worth playing? Explain your answer.

Expected winnings each game = £40 x 1/20 = £2

But cost of game is only £1, so you can expect to win money in the long run

Experimental probabilitySometimes the probability of an event occuring is not understood (eg trying to predict the stock market!) very well. Experimental data can be collected to give an estimate of the actual probability.

If an event occurs x times in n trials, the probability of the event is approximated by x/n

Eg Bob is convinced his toast always lands butter side down when he drops it. He drops a piece of toast and it lands butter side down 30 times in 50 attempts.Comment on Bob’s claim.

so probability

x = 30 and n =50

53

5030

Bob’s claim is supported by the data, although he has not conducted that many trials so it is possible he was just unlucky.

The more trials, the more likely it is that the experimental data matches the actual theory

Eg Bob repeats the experiment, dropping the piece of toast 1000 times.It lands butter side down 600 times. Comment on Bob’s claim now.

so probability

x = 600 and n =1000

53

1000600

Bob’s claim is more strongly supported by the data, as it very unlikely he would be ‘unlucky’ that many times.

Eg A bag contains some red counters and blue counters.

There are n red counters. There is 1 more blue counter than red counters.Bob will take a counter at random from the bag, record the colour and pick another. The probability that Bob picks two red counters is 1/6.

Prove that

)( 1 nnred + blue = total

12 nP(1st pick red) =

12 nn

So P(both red) =n

n

n

n

2

1

12

)(

)(

122

1

nn

nn

6

1 2416 nn )(

4n

P(2nd pick red) =n

n

2

1

24

1

n

n

But P(both red)6

1

24

1

n

n

2466 nn

82 n

4 n

, giving

Problem solving

1. A bag contains some black counters and white counters.

There are n black counters. There are 2 less white counters than black counters.Bob will take a counter at random from the bag, record the colour and replaces it before picking another. The probability that Bob picks one of each counter is 3/8.Find how many of each coloured counter there are in the bag

)( 2 nnred + blue = total

22 n

P(BW) =22

2

22

n

n

n

n222

2

)(

)(

n

nn

Problem solving

P(WB) =2222

2

n

n

n

n222

2

)(

)(

n

nn

But P(one each)8

3

2223216 )()( nnn

1224123216 22 nnnn

01284 2 nn

0322 nn

013 ))(( nn

3 n as n > 0So P(one each)222

22

)(

)(

n

nn

So 3 black, 1 white counter in bag

2. Gary plays two games of chess against Mijan.The probability that Gary will win any game against Mijan is 0.55The probability that Gary will draw any game against Mijan is 0.3

In a game of chess, you score 1 point for a win, ½ point for a draw, 0 points for a loss.

Work out the probability that after two games, Gary’s total score will be the same as Mijan’s total score.

P(Gary loses) = 0.15

Total scores are the same if:

Mijan wins 1st game, Gary wins 2nd

Gary wins 1st game, Mijan wins 2nd

Both games drawn

08250150550 ...

08250550150 ...

0903030 ...

2550.

(4 marks)

The types of people watching a film at a cinema are shown in the table.

Two of these people are chosen at random to receive free cinema tickets.

Calculate the probability that the two people are adults of the same gender.

P(Adult Male, Adult Male) P(Adult Female, Adult Female)+

TOTAL = 50

2150

2049

x +1450

1349

x 635

+ 13175

43175

Misconceptions

If you toss a fair coin and get heads 5 times in a row, you are more likely to get tails the next time.

In a football match, you can either win, lose or draw. So the probability of winning is 1/3.

You are less likely to win with lottery numbers 1,2,3,4,5,6 than if you pick numbers at random

If you toss a coin 50 times and get heads 40 times, the coin must be biased

If you roll two dice and add the results, the probability of getting 9 is 1/11 as there are 11 possibilities (2-12)

When tossing a coin, you are just as likely to get 5 heads in a row as 10 in a row- it’s just chance

Discuss why each statement is incorrect

The probability is the same each time- previous results are irrelevant

Winning may not have the same probability as losing

Every number has the same chance and so does every combination

It might be biased, as you would only expect 25 heads, but it is still possible to get 40 out of 50 heads with a fair coin.

There are more ways to get some totals than others

P(5 heads in a row) = 1/32

P(10 heads in a row) = 1/1024