Chapter Probability - St Leonard's College...experimental probability of certain disasters,...

Chapter

Review of probability

(Consolidating)

Unions and intersections

The addition rule

Conditional probability

Two-step experiments

using tables

Using tree diagrams

Independent events

8A

8B8C8D8E

8F8G

S T A T I S T I C S A N D P R O B A B I L I T Y

Chance

Describe the results of two- and three-step chance experiments, both with and without replacement, assign probabilities to outcomes and determine probabilities of events. Investigate the concept of independence.Use the language of ‘if . . . then’, ‘given’, ‘of’, ‘knowing that’ to investigate conditional statements, and identify common mistakes when interpreting such language.(10A) Investigate reports of studies in digital media and elsewhere for information on the planning and implementation of such studies, and the reporting of variability.

What you will learn

Australian curriculum

8 Probability

Essential Mathematics for the Australian Curriculum Year 10 2ed

ISBN 978-1-107-56890-7 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.

Cambridge University Press

We have all heard about or experienced a disaster of some kind in Australia. Floods, bushfires or cyclones, for example, are very real possibilities in many parts of the country and, because of this, most people take out insurance to cover for any potential losses. Consequently, insurance companies try to predict the likelihood of such disasters and set their premiums

to suit. This is called risk analysis and includes calculating the experimental probability of certain disasters, depending on a number of factors. Some factors include:• geographic location• historical records of previous

disasters• local facilities, such as hospitals

and power supply• area topography.

• Chapter pre-test• Videos of all worked

examples• Interactive widgets• Interactive walkthroughs• Downloadable HOTsheets• Access to HOTmaths

Australian Curriculum courses

Online resources

Disaster probability




• • • • •

••

•••

0.10 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

impossible unlikely evenchance

likely certain

•




4 1

23

4 1

1

23

34

4 11

2

23

3

1

1

5

4

4

4

33

4

4

5

1

759

311




10 cm

A

B

C

10 cm 10 cm




•••••

•• ξ

•

A

A′

A B

A′• ∅• ∩• ∪•

∩ ∅• ′

• ′

• ∩ ′

•




1

A B

n(A′ ∩ B)(B only)

n(A ∩ B′)(A only)

n(A′ ∩ B ′)(neither A nor B)

n(A ∩ B)

43

2

1

3

4

2

4 6

5

5

10

A

n(A ∩ B)

n(A ∩ B′)n(A′ ∩ B)

n(A′ ∩ B′)

n(B)

n(B′)n (ξ)

n(A) n(A′)

A′

B′

B

∩ ∪

∩ ∪

A B

2 2

2

4

∩

′ ∩ ′

∩ ∩∪ ∪

∩ ∩

∪ ∪

∩ ∅∩




6

C N

86

10

∩

1

A B

62

3

∩ ′ ∩ ∩ ′′ ∩ ′ ′

∪

∩ ′ ∩ ′




′

′

′

∩ ′ ∩′ ∩ ′ ′ ∩ ′ ′

′ ξ

∩ ∩′ ∩ ′ ∩∩ ′ ∩ ′′ ∩ ′ ′ ∩ ′

∩ ′ ∩′ ′ ∩ ′ ′ ∩ ′

∪ ∪ ∩ ∩ ′ ′ ∩

∩

′

∩ ′

∩

A B A B A B

∪

A B A B A B

′ ∪ ′

A B A B A B




∪ ∩ ∅

. . .

. . .

∩ ∪

∩ ∪

∩ ∪




∩ ∪′

A

5 6

3

2

B

∩ ′ ∩ ∩ ′ ′ ∩ ′′ ∪ ξ

∩ ′ ∩ ′

′ ∩ ′ ∩ ′ ∩ ∪




′

′

′

′

∪ ′ ′ ∩ ′




L S

I




••••••

•




A

6 4 5

B

∪•• ∩ ∩• ∪ ∩•

∪ ∩

A B

A ∪ B

A

A

B A B

B

A B

A ∩ B

• ∩• ∪

A B




∩

′ ∩

∪

∩

′

∩

′

∪

∩

∩∪

∪ ′′ ∩ ′




∪

∩ ′ ∩ ′

∪ ∩

∩

∩

∴ ∩

∩

′ ∩ ′ ′ ∩ ′ ∪ ′

A B

••

∪∩

∪

∪

∩∩∩∩

∩ ∪∪ ∩




∩

′ ∩

∪

. . .

∩ ∪

∪∩ ′ ∩ ′

∪∩ ′ ∩ ′




∩ ′ ∪′ ∩ ∪

∪

∩∪

∪ ∪

A

B C




••

••••

∩ ∩

∩ ∩

A B

Pr(A)

A B

Pr(A|B)




∩

2

A B

43

0

∩

′

′

∩

∩ ′




∩

∩

∩∩ 2

A B

75

′

′

∩∩

∩

3

A B

46 4

A B

310

5

A B

21 2

A B

65

3




∩′

′

′

′

′

′

′

′




′ ∩ ′

′

′

′

′

A

8 2B

∩∩

∩




∩ ′ ∩ ′

∩ ′ ∪

A

13 7 8

5

B

∩∪ ′ ∩ ′

∩

A

10 5 15

0

B




••

32 14 25

22 53

31 34 42not allowedwithout replacement

••




∩




R

R

B

B

Pr (R, B) = 2_9

R

R

R

R

B

R

R

B

(R, R) 1_91_91_91_91_91_91_91_91_9

(R, R)

(R, B)

(R, R)

(R, R)

(R, B)

(B, R)

(B, R)

(B, B)

Choice 1 Choice 2 Outcome Probability

2_3

2_3

2_31_

3

1_3

1_3

(R, R)

(R, B)

R

B

(B, R)

(B, B)

Pr (R, B) = 2_9

2_3

2_3

4_9

× =

2_3

1_3

2_9

× =

1_3

1_3

1_9

× =

2_3

1_3

2_9

× =

R

B

B

Choice 1 Choice 2 Outcome Probability

R

•∩

•∩

∩

2_3

2_3

2_3

1_3

1_3

1_3

R⎪R

B⎪R

R ∩ R

R ∩ B

R⎪B

B⎪B

B ∩ R

B ∩ B

R

B




•

•

•

•••




1_2

1_2

1_2

1_2

1_4

3_4

red

Counter Outcome ProbabilityBox

green

(A, red)

(A, green)

red

green

(B, red)

(B, green)

1_2

1_2

1_4

× =

1_2

1_2

1_4

× =

1_2

3_4

3_8

× =

1_4

1_2

1_8

× =

A

B

5_8

4_7

3_7

3_8

5_7

2_7

B

Selection 2 Outcome ProbabilitySelection 1

W

(B, B)

(B, W)

B

W

(W, B)

(W, W)

5_8

4_7

__14

× =

5_8

3_7

× =

3_8

2_7

× =

5_7

3_8

× =

B

W

5

__5615

__5615

__283




3_8

3_8 (M, M)

(M, F) ( _________ )

( __, __ ) ( _________ )

( __, __ ) ( _________ )

3_8

3_8

__64

× =

5_8

MM

F

M

F

F

9( )

5_8

3_7

MM

F

M

F

F

(M, M) ( _________ )

( __, __ ) ( _________ )

( __, __ ) ( _________ )

( __, __ ) ( _________ )

1_2

1_4 (A, yellow)

__________

__________

________________

________

________

________

1_2

1_4

1_8

× =A

yellow

orange

yellowB

Counter Outcome ProbabilityBox




__________

__________

__________

_____________

_____

_____

__________ ________

________

________

Falconwhite

Commodore

Colour Outcome ProbabilityCar

1_2

1_2




A: 9 pointsB: 7 points

A winsA: 10 pointsB: 7 points

B winsA: 9 pointsB: 8 points


Pr(A winning)

Pr(B winning)




1_2

1_4

1_8

1_8




2

A B

35

2_5

2_5

2_5

3_5

3_5

3_5

A⎪A

B⎪A

A ∩ A

A ∩ B

A⎪B

B⎪B

B ∩ A

B ∩ B

A

B

2_5

1_4

2_4

3_5

3_4

2_4

A⎪A

B⎪A

A ∩ A

A ∩ B

A⎪B

B⎪B

B ∩ A

B ∩ B

A

B

∗∩ ∩

∩

•

•

•

1

N T

45




•• ∩

1

A B

43

2




2

A B

14

3




1

A B

12 2

A B

65

15

3

A B

62

4

A B

2 5

′

′

′

′

′

′

′

′




A

2

B

4

3

1

A B

32

x

1

A B

211

x

7

x

2

B

14

A

1

A B

2x

4

∩

∩

∩ ∪

∪




12

34

5

67




Up for a challenge?If you get stuck on a question,check out the 'Working withunfamiliar problems' poster

at the end of the bookto help you.

∪

chee

seEx

it 1

cheese

Exit 3

cheese

Exit 2 Exit 4

Entrancestarting point

nocheese

Exit 5 cheese




Addition rule

If A and B are mutually exclusive:Pr(A ∩ B ) = 0 andPr(A ∪ B ) = Pr(A ) + Pr(B )

Independent events

- Pr(A⎪B ) = Pr(A)

- Pr(A ∩ B ) = Pr(A ) × Pr(B )

Pr(A ∪ B ) = Pr(A ) + Pr(B ) – Pr(A ∩ B )

Unions and intersections

• Union A ∪ B (A or B )

• Intersection A ∩ B (A and B )

• Complement of A is A′′ (not A)

• A only is A ∩ B ′

• Mutually exclusive events A ∩ B =

A B

A B

A B

A B∅

A A′

Review

Sample space is the list of all possible outcomes.

Pr(event) = number of favourable outcomestotal number of outcomes

Venn diagram Two-way table

2

A B

54

1

246 6 12

51

75

Conditional probability

Pr(A⎪B ) = orPr(A ∩ B )Pr(B)

n(A ∩ B )n(B)

Pr (A⎪B ) = 27

2

A B

54

1

24

51

75

6 6 12

With replacement Without replacement

AABC

(A, A)(A, B)(A, C)

(B, A)(B, B)(B, C)

(C, A)(C, B)(C, C)

B C AABC

×(A, B)(A, C)

(B, A)×

(B, C)

(C, A)(C, B)

×

B C

Tree diagrams

With replacementChoice 1 Choice 2 Outcome Probability

Without replacement

(W, W)

(W, B)

(B, W)

(B, B)

3_7

4_7

W

B

4_7

4_7

3_7

3_7

W

B

W

B

__499

__4912

__4912

__4916

(W, W)

(W, B)

(B, W)

(B, B)

3_7

4_7

W

B

4_6

3_6

2_6

3_6

W

B

W

B

__71

__72

__72

__72

3 white4 black

Tables

Pr (1 white, 1 black) = 2_7

2_7

4_7

+

=

Probability

•

•

A A′BB′

A A′BB′

••




∩ ∅∩ ∪

′

∩ ′′

′




5

A B

23

1

′ ∅ ∪




∩

′ ∩∪

∪∩ ′ ∩ ′

2

A B

35

4

′

′

1

A B

x3

3

∩ ∪




Chapter Probability - St Leonard's College...experimental probability of certain disasters,...

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