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Probability

Probability

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Probability

1SERIES TOPIC

J 15

Probability measures the chance of something happening. This means we can use mathematics to find how likely it is that an event will happen.

PROBABILITY

What do I know now that I didn't know before?

Answer these questions, before working through the chapter.

I used to think:

Answer these questions, after working through the chapter.

But now I think:

What is an outcome?

What is the range of probability from impossible to certain?

What is the probability of rolling a 6 with a single die?

What is an outcome?

What is the range of probability from impossible to certain?

What is the probability of rolling a 6 with a single die?


Probability

2SERIES TOPIC

J 15

Basics

Let's say a bag contains 4 red stones and 4 blue stones. One of the stones is pulled out

So relative frequency can change from experiment to experiment. Here is an example:

Basic Definitions

Here are four basic terms for probability:

Relative Frequency

Each outcome has a relative frequency where

A bag is full of 10 red, 10 blue and 10 green stones. Let's say 12 stones are drawn from the bag at random in this order:

The next time 14 stones are drawn, they are drawn in this order:

The relative frequency of red stones is or124

31 .

The relative frequency of blue stones is or123

41 .

The relative frequency of green stones is 125 .

The relative frequency of red stones is or146

73 .

The relative frequency of blue stones is 145 .

The relative frequency of green stones is 143 .

R G G B G G B B R R R G

B B G R R B R R B R B R G G

• Event – An event is a situation which could have different outcomes.

• Outcome – An outcome is a possible result of an event.

• Sample Space – All the possible outcomes. For example, the same space of rolling a die is {1, 2, 3, 4, 5, 6}

and the sample space of tossing a coin is {H , T}.

• Random Event – An event with equally likely outcomes.

• The event is that a stone is selected from the bag.

• The possible outcomes are that a red stone or a blue stone could be selected. So the sample space is {R , B}.

• The event is a random event because there are the same number of red stones as blue stones. If there was only one blue stone, then the event would not be random because it is more likely that a red stone is drawn.

Relative frequency total number of trialsnumber of times outcome occurs=

a

b

Relative frequency is also sometimes called "experimental probability".


Probability

3SERIES TOPIC

J 15

Questions Basics

What is the sample space (the possible outcomes)?

What is the sample space of rolling a die and is rolling a die a random event? Why?

What is the relative frequency of broken boxes?

What is the relative frequency of unbroken boxes?

If the box factory produced 3600 boxes, what number of these boxes is likely to be broken?

In this experiment, what is the relevant frequency of a 2 being rolled?

In this experiement, what is the relevant frequency of a 4 being rolled?

Is this a random event? Why?

1. A coin is tossed once.

2. A single die is rolled ten times and these are the numbers it shows: 4, 5, 1, 4, 2, 6, 1, 3, 4, 1

3. A box factory noticed that 6 out of 72 boxes were broken.

a

a

a

b

c

b

c

b


Probability

4SERIES TOPIC

J 15

BasicsQuestions

Fruit Number boughtApples 81

Peaches 78

Apricots 84

Oranges 75

Bananas 82

How many items of fruit were bought in total?

What is the relative frequency of apricots?

What is the relative frequency of oranges?

The owner notices that 3 of the apples he bought were rotten. What is the frequency of rotten apples within the apples?

If the owner bought 360 apples over a week, how many apples should they expect to be rotten?

4. A fruit shop buys fruit according to this table:

a

b

c

d

e


Probability

5SERIES TOPIC

J 15

Knowing More

The formula for probability could also be used in other ways.

How many possible outcomes are there?

What is the probability that a 4 is rolled?

What is the probability that an even number is rolled?

What is the probability that a number greater than 2 is rolled?

What is the relative frequency of faulty watches?

If the watchmaker made 1500 watches in total, how many would he expect are faulty?

6 (the die could roll a 1, 2, 3, 4, 5 or 6)

a

b

c

d

a

b

Probability

A bag contains 1 red stone, 1 blue stone and 1 green stone. If you choose a stone without looking, what are the chances you will choose a red stone? The probability would be

31 .

This is the formula to find the probability of an outcome "X":

The 'total number of outcomes' is also the size of the sample space. Here is an example:

total number of outcomesnumber of ways could occur

XX

P =^ h

A single die is rolled, answer the following questions

A watchmaker selects 100 of his watches at random and notices 6 of them are faulty.

total number of outcomesnumber of ways to roll a

or .

P 44

61 0 16

=

= o

^ h

total number of outcomesnumber of even rolls

or

even

.

P63

21 0 5

= =

=

^ h

total number of outcomesnumber of rolls greater than

or .

P 22

64

32 0 6

2 = =

= o

^ h

number of faulty watches100 100

6=

total number of watchesnumber of faulty watches

number of faulty watches

faultyP

1006

1500

=

=

^ h

number of faulty watches 1500 901006#= =

This will not definitely happen. It is just an expected value based on the probability.


Probability

6SERIES TOPIC

J 15

Knowing More

Probability is Always Between 0 and 1

The number of ways any outcome X can occur will always be less than or equal to the total number of outcomes. So the probability will always be a fraction. This means that:

Complementary Probability

The complement of an outcome is when the outcome doesn't occur.

0 1XP# #^ h

• If 0XP =^ h it means that the outcome X is impossible. If ,1XP =^ h it means that the outcome X is certain.

• The closer XP^ h is to 0, the more unlikely X is and the closer XP^ h is to 1 the more likely X is.

• If you find a probability greater than 1 then a mistake has been made somewhere.

• The notation Xu ( u is called tilde) means the complement of X and so XP u^ h means the probability of X not occuring.

• The total probabilities of the outcomes must add up to 1 and so 1X XP P+ =u^ ^h h . This means we can use the formula:

1X XP P= -u^ ^h h

A bag contains 4 blue stones, 3 green stones and 5 red stones. Answer these questions if one stone is drawn at random

Find the probability that the stone will not be blue.

Find the probability that the stone will not be red.

Find the probability that the stone will not be green.

a

b

c

not blue 1 blueP P

1124

32

= -

= -

=

^ ^h h

not red 1 redP P

1125

127

= -

= -

=

^ ^h h

not green 1 greenP P

1123

43

= -

= -

=

^ ^h h


Probability

7SERIES TOPIC

J 15

Questions Knowing More

What is the probability it opens on page 89?

What is the probability it is opened to an odd page?

What is the probability of opening to page 65 or after?

What is the probability of opening to a page after page 65?

What is the probability that it is not opened to page 30?

What is the probability that it is opened to a page number which is a multiple of 5?

What is the probability that the page is not a multiple of 5?

How are the answers from f and g related? Why?

1. A book has 120 pages and is opened to a random page.

a

b

c

d

e

f

g

h


Probability

8SERIES TOPIC

J 15

Knowing MoreQuestions

What is the experimental probability of incorrect answers based on this information?

If a card is drawn at random, what is the size of the sample space?

If a card is drawn at random, what is the probability it is an ace?

If a card is drawn at random, what is the probability it is the ace of spades?

If a card is drawn at random, what is the probability it is not a diamond?

If a card is drawn at random, what is the probability it is red?

Based on this information, how many questions would you expect are incorrect if the test totalled 180 questions?

2. You notice that when taking a test, you get 1 out of the first 12 questions incorrect.

3. A standard deck of cards (with no jokers) is shuffled and placed face down and spread out.

a

a

b

c

d

e

b


Probability

9SERIES TOPIC

J 15

Questions Knowing More

The probability of selecting a single history book is 92 . How many history books would you expect to be in a

group of 36 randomly selected book?

The probability of selecting a single book about food is 94 . What is the probability of selecting a single book

about sport?

If 54 books are selected at random, how many of each type of book could be expected to be selected?

4. A library has books about sport, food and history only.

a

b

c


Probability

10SERIES TOPIC

J 15

Knowing MoreQuestions

What is the sample space of this experiment? (Hint: How many different letters are there?)

Which letters have the greatest chance of being selected? Why?

What is the probability of selecting an 's'?

What is the probability of not selecting the 'M'?

5. The letters from the word Mississippi are put into a bag. Answer the following questions if a single letter is drawn:

a

b

c

d


Probability

11SERIES TOPIC

J 15

Using Our Knowledge

Mutually Exclusive Events

Mutually exclusive events are events that cannot occur at the same time. If they are not mutually exclusive they are called inclusive events.

For example, rolling a 2 and rolling a 3 with a single die are mutually exclusive events – they can't happen at the same time.

If X and Y are mutually exclusive events then or Y YX XP P P= +^ ^ ^h h h.

Here is an example comparing mututally exclusive events and inclusive events:

X: The page number is a multiple of 5 X: The page number is a multiple of 10

Y: The page number is even Y: The page number is odd

These are inclusive events because 10 and 20 are a multiples of 5 and even which means X and Y can happen at the same.

These are mutually exclusive events because there are no numbers which are multiples of 10 and odd. So X and Y can't happen at the same time.

In the Venn diagram above it's easy to see that 10 and 20 are both multiples of 5 AND even numbers. The sets overlap.

In the Venn diagram above it's easy to see that the sets of multiples of 10 and odd numbers are mutually exclusive. The sets do not overlap and are separate.

Find the probability of rolling a 2 or a 3 with a single die

A book has 20 pages and is opened to a random page. Which of these are mutually exclusive?

or2 3 2 3P P P

61

61

31

= +

= +

=

^ ^ ^h h h

orX Y X YP P P` ! +^ ^ ^h h h or

.

X Y X YP P P

202

2010

53 0 6

` = +

= +

= =

^ ^ ^h h h

7 5

1

9

3

13

15 10

18

1

9

17

11

13 1910 20

12

6 4 2

8

18 16

1514

3

7514

1216

8

6

2

420

1117

19

Multiples of 5

Odd Numbers

Multiples of 10

Even numbers

Numbers from 1 – 20 Numbers from 1 – 20


Probability

12SERIES TOPIC

J 15

Using Our Knowledge

If X and Y are inclusive then there is an extra step to find orX YP^ h. Subtract the probability of the 'overlapping' outcomes. To continue the example from the previous page.

X: The page number is a multiple of 5

A compound event involves more than one outcome. It could have two stages or more. To find the probability of compound events, find the probability of each outcome and multiply them together.

Step 1: Find the probability that the first stone is blue.

Step 2: Find the probability that the second stone is blue.

Step 3: Multiply the probabilities together:

Y: The page number is even

A book has 20 pages and is opened to a random page. Find the probability that the page number is even OR a multiple of 5

or andX Y X Y X YP P P P= + -^ ^ ^ ^h h h h

.

204

2010

202

2012

53 0 6

= + -

=

= =Since some page numbers are in both X and Y they shouldn't be counted twice

7 5 20

10

18

8

6

1614

12

2

4

151

9

3

13

11

19

17

Multiples of 5

Even numbers

Numbers from 1 – 20

Compound Events

A bag holds 6 red stones and 4 blue stones. Find the probability of drawing two blue stones from two draws.

total stonesnumber of blue stones1st stone is blueP

104

52

=

=

=

^ h

total stones remainingnumber of blue stones remaining

2nd stone is blueP

93

31

=

=

=

^ hThere is one less blue stone from the previous draw

There is one less stone in the bag from the previous

2 blue stones 1st stone is blue 2nd stone is blueP P P

52

31

152

#

#

=

=

=

^ ^ ^h h h


Probability

13SERIES TOPIC

J 15

Using Our Knowledge

G

G

G

G

G

Tree diagrams are used with compound events to see all the possible outcomes (the sample space).

Tree Diagrams

A bag contains 2 red, 1 green and 2 blue stone.

What are all the possible methods to select two stones?

From the tree diagram we can that there are 8 possible outcomes in this sample space . {RR, RB, RG, BR, BB, BG, GR, GB} (GG is not in the sample space because there is 1 green stone.)

What is the probability that a blue stone is selected first and a red stone selected second? (without replacing the blue stone)

What is the probability of drawing one red and one green stone?

There are two possible outcomes with one red stone and one green table: GR and RG

Notice the difference between b and c . In b , the order matters so there is only one way to draw blue first and red second. In c , the order doesn't matter, so both GR and RG are counted.

a

b

c

First draw Second draw Sample space

Sample space(all possible outcomes)

BRP52

42

51#= =^ h

orGR RG GR RGP P P

51

42

52

41

204

51

# #

= +

= +

=

=

^ ^ ^

` `

h h h

j j

R

R

R

R

B

B

B

B

G

G

R

R

R

B

B

B

R

R

R

B

B

B

52

41

41

41

41

42

42

42

42

52

51


Probability

14SERIES TOPIC

J 15

Using Our Knowledge

If the compound event is just a two-stage event, then a two-way table can be used.

Tables for Two-Stage Events

Two multiple choice questions with options A, B and C need to be answered.

A restaurant serves 5 starters, 4 mains and 3 desserts. How many ways are there to order a three course meal of a starter, main, and dessert?

How many possible ways are there to answer the questions?

What is the probability that both answers are A?

What is the probability that both answers are the same?

What are the chanced that the answers are different?

Only one outcome has both answers as A.

There are 9 possible ways to answer the Q1 and Q2: {AA, BA, CA, AB, BB, CB, AC, BC, CC}. So the sample space size is 9.

To find the sample size – of a compound event – without a table or tree diagram, multiply the sample sizes of each stage. In the example above there are 3 ways to answer Q1 and 3 ways to answer Q2, so the sample size is 3 3 9# = .

a

b

c

d

Q1

A B CQ2

A AA BA CA

B AB BB CB

C AC BC CC

both answers are A AAP P

91

` =

=

^ ^h h

or orsame answers AA BB CC

AA BB CC

P P

P P P

91

91

91

31

=

= + +

= + +

=

^ ^^ ^ ^

h hh h h

different answers 1 same answersP P

131

32

= -

= -

=

^ ^h h

starters mains dessertsdifferent ways to order a three coursemeal

5 4 3

60 .

# # # #=

=


Probability

15SERIES TOPIC

J 15

Questions Using Our Knowledge

2. Identify in the following if outcomes A and B are mutually exclusive or not. Give a reason why you say so.

3. A single die is rolled. Answer the questions about these outcomes (check if they are mutually exclusive first):

A: Obtaining 'heads' from a coin toss

B: Obtaining 'tails' from a coin toss

A: Rolling a 1 or a 6

B: Rolling an even number

Find P (B or D).

Find P (A or C).

Find P (A or D).

Find P (B or C).

A: Finishing a task between Monday and Thursday

B: Finishing a task between Saturday and Tuesday

1. What is the difference between mutually exclusive events and inclusive events?

a

a

c

b

d

b

C: Rolling an odd number

D: Rolling a 3


Probability

16SERIES TOPIC

J 15

Using Our KnowledgeQuestions

4. Use this information to answer the following questions:

Are A and B mutually exclusive? Are C and D mutually exclusive?

Are B and C mutually exclusive?

Find P (B and C).

a b

c

e

• P A21=^ h

• P B103=^ h

• P orA B54=^ h

• P orC D10027=^ h

• P orB C2511=^ h

• P C51=^ h

• P D252=^ h

If P (A or C) = 53 ,

use P (A or C) = P (A) + P (C) - P (A and C) to find P (A and C).

d


Probability

17SERIES TOPIC

J 15


5. A book with 50 pages is opened up to a random page. Answer questions that follow about these outcomes:

Find P (A or B).

Find P (B or E).

Find P (E or F).

Find P (A or F).

Find P (C or D).

Find P (B or D).

• A: The page number is a multiple of 10

• B: The page number has a 7 in it

• C: The page number is a multiple of 4

a

c

e

b

d

f

• D: The page number is a multiple of 3

• E: The page number is 20 or below

• F: The page number is 45 or more

(Hint: First check if they are mutually exclusive)


Probability

18SERIES TOPIC

J 15


6. A single die is rolled twice.

What is the probability of rolling a 5 and then a 1?

What is the probability that both rolls will be greater than 2?

What is the probability that both rolls will be even?

What is the probability of both rolls being 6?

What is the probability that both rolls will be a 2 or a 4?

a

b

c

d

e


Probability

19SERIES TOPIC

J 15


Y

Y

O

Y

W

O

Y - Y

Y - W

W - O

-

-

-

-

-

-

7. A bag contains 1 yellow, 1 white and 1 orange stone. A stone is drawn at random and then replaced. Then a stone is drawn at random for a second time.

Complete the tree diagram below for this compound event:

How big is the sample space of this experiment? Is this what you expected?

What is the probability the white stone will be drawn first?

What is the probability the white stone will be drawn second?

What is the probability that the white stone will be drawn both times?

a

b

c

d

e



Probability

20SERIES TOPIC

J 15


Redraw the tree diagram if the stone that is drawn first is not replaced?

What is the size of the sample space now?

What is the probability that the yellow stone will be drawn first?

What is the probability that the yellow stone will be drawn second?

What is the probability that the yellow stone is drawn both times?

f

g

h

i

j


Probability

21SERIES TOPIC

J 15


8. A tennis tournament has a singles trophy and a doubles trophy. The countries competing for the singles trophy are: India, Spain and Greece. The countries competing for the doubles trophy are just India and Spain. Each country has equal chance to win the trophies.

Draw a table for this two-stage event of trophy winners.

What is the probability that Greece will win the singles trophy?

What is the probability that India will win both trophies?

What is the probability that India and Spain will win a trophy each?

What is the probability that Spain and Greece will win a trophy each?

What is the probability that India will not win a trophy?

a

b

c

d

e

f


Probability

22SERIES TOPIC

J 15

Thinking More

When probabilities of each outcome are written on the 'branches' of a tree diagram it is called a probability tree diagram. To find the probability of each multi-stage event just multiply the probabilities of each branch.

Probability Tree Diagrams

The probability of flipping a heads with a trick coin is 54

Draw a tree diagram representing 3 flips of this coin:

Find the probability of flipping 3 heads?

Find the probability of flipping T-H-T.

a

b

c

HHHP54

54

54

12564

# #=

=

^ h

THTP51

54

51

1254

# #=

=

^ h Multiply the probabilities of each branch

Multiply the probabilities of each branch

H

T

H

T

H H

H

H

H

H

TTH

T

H

T

H

T

T

H

54

54

54

b

c

51

51

54

51

51

54

51

54

51

54

51

#

#

#

#

H T H

H T T

T H H

T H T

T T H

T T T


Probability

23SERIES TOPIC

J 15

Thinking More

Venn diagrams help us see groups better. Let's say that from 30 students, 12 play football, 11 play tennis and 5 play both then the Venn diagram would look like this:

So the diagram shows that 7 students play football only, 6 students play tennis only, 5 students play both and 12 students don't play either game.

• The part in both circles is called the intersection. It has this symbol "+". In the example above there are 12 (7 + 5) students in football, 11 (6 + 5) in tennis and 5 in the intersection football + tennis. This means that 5 students played football AND tennis.

• The union of sets is the combined set and has the symbol ",". Above example football , tennis has 7 + 5 + 6 = 18 students. This means that students play football OR tennis.

Probability from Venn Diagrams

Football

12

7 5 6

Tennis

Find the probability that the student plays football.

Find the probability that the student plays tennis or football.

Find the probability that the student plays tennis and football.

a

b

c

Let's say a student is selected at random from the group represented in the above Venn diagram

numbers of studentsnumber of football players

footballP

307 5

52

=

= +

=

^ h

numbers of studentsnumber of students who play football or tennis

football tennisP

307 5 6

53

, =

= + +

=

^ h

numbers of studentsnumber of students who play football and tennis

football tennisP

305

61

+ =

=

=

^ h


Probability

24SERIES TOPIC

J 15

Thinking More

Sometimes tables are used in probability questions.

Probability from Tables

If a medal is selected at random then what is the probability it is gold?

If a silver medal is selected at random then what is the probability it would be for archery?

If a medal from fencing is selected at random, what is the probability it is bronze?

If a medal is selected at random, what is the probability it will be a gold medal for swimming?

The type of table in the above example is called a 'contingency' table.

a

b

c

d

At the olympics, a country won medals for events according to this table

Gold Silver Bronze TotalSwimming 11 15 7 33

Fencing 10 13 5 28

Archery 16 11 12 39

Total 37 39 24 100

totalmedalsnumber of goldmedals

gold medalP

10037

=

=

^ h

total silver medalsnumber of silver medal for archery

silver medal for archeryP

3911

=

=

^ h

total fencingmedalsnumber of bronze fencingmedals

fencing medal is bronzeP

285

=

=

^ h

totalmedalsnumber of goldmedals for swimming

gold medal from swimmingP

10011

=

=

^ h


Probability

25SERIES TOPIC

J 15

Questions Thinking More

1. What do the symbols + and , mean?

2. A group of 20 people are asked which pets they have. This is represented in this Venn Diagram:

How many people have dogs only?

How many people have dogs and cats?

If a person is chosen at random find the probability that they have cats only, P (cats).

Find the probability of a person having cats and dogs, P (cats + dogs).

Find the P (cats , dogs).

a

b

c

d

e

Cats Dogs

7 94


Probability

26SERIES TOPIC

J 15

Thinking MoreQuestions

Cats Dogs

Fish

66

3

5

37

10

3. A different group of people was asked about which pets they had and this is the resulting Venn diagram:

How many people have fish only?

How many are in the set cats + dogs?

How many people in the set dogs + fish?

How many people have all 3 pets?

If a person is selected at random, then find P (cats + dogs).

If a person is selected at random, then find P (cats , dogs).

If a person is selected at random, then find P (dogs + fish).

a

b

c

d

e

f

g


Probability

27SERIES TOPIC

J 15

Questions Thinking More

4. A probablity tree diagram for a certain compound event looks like this:

How many stages are in this compound event?

Use the tree diagram to find P (BGU).

Use the tree diagram to find P (AEQ).

a

Use the tree diagram to find P (ADN).e

c

d

Use the tree diagram to find P (AD).b

I

K

J

L

M M

N

O

P

Q

R

S

T

U

V

W

H

C

A

B

D

E

F

G

X

Y

Z

52 5

1

51

106

203

207

53

21

92

403

52

51

51

31

2013

101

81

94

4011

21

21

83

101

101

54

53

A C I

A C J

A C K

A D L

A D

A D N

A E O

A E P

A E Q

B F R

B F S

B F T

B G U

B G V

B G W

B H X

B H Y

B H Z


Probability

28SERIES TOPIC

J 15

Thinking MoreQuestions

5. The table below is of a group of teenagers who play different sports.

Tennis Football Volleyball TotalBoys 11 5 8 24

Girls 9 11 6 26

Total 20 16 14 50

If a student is selected at random what is the probability of selecting are a boy?

If a student is selected at random what is the probability of selecting a volleyball player?

If a student is selected at random what is the probability of selecting a girl who plays football?

If a boy is selected at random what is the probabiliy he plays football?

If a football player is selected at random, what is the probability they are a girl?

a

b

c

d

e


Probability

29SERIES TOPIC

J 15

Answers

1.

1.

1.

2.

2.

2.

3.

4.

5.

3.

4.

Basics:

Knowing More:

Knowing More:

Using Our Knowledge:

a

a

a

a

a

b

b

c

b

b

This is a random event because each outcome is equally likely.

The sample space of rolling a dice is {1, 2, 3, 4, 5, 6} and it is a random event because each outcome is equally likely.

Relative frequency1211=

Heads and tails or {H,T}

Relative frequency of a 2 being rolled is 101 .

Relative frequency of a 4 being rolled is 103 .

Relative frequency of broken boxes is 121 .

c 300

400a

c

d

e

Relative frequency of apricots is 10021 .

Relative frequency of oranges is 163 .

Relative frequency of rotten apples within

the apples is 271 .

Apples expected to be rotten = 13.3333...

21b c

d e

1201

157

2411

120119

g

h

54

When you add the two probabilities, they equal 1. This is because these are all the possible outcomes, and all the possible outcomes have a probability of 1 because a probability of 1 covers all outcomes.

b 15

a

a

a

The size of the sample space is 52.

f51

121

c

d e

b131

521

21

43

8

d

'i'and 's' as they appear 4 times each in the word Mississippi.

Probability of selecting an 's' is 114 .

b

b

c

c

93

{m, i, s, p}

1110

Food: 24

History: 12

Sport: 18

a

b

Mutually exclusive events cannot occur at the same time.

Mutually inclusive events can occur at the same time.

Rolling 'heads' and 'tails' in a coin toss are mutually exclusive events. This is because these cannot be done at the same time.

Finishing a task between Monday and Thursday and finishing a task between Saturday and Tuesday are mutually inclusive events. This is because they can be done at the same time. The task could be completed on Monday or Tuesday.


Probability

30SERIES TOPIC

J 15

Answers

Using Our Knowledge: Using Our Knowledge:

3. 7.

4.

6.

P (B or D) 32=

P (A or C) 32=

P (A or D) 21=

P (B or C) 1=

a

c

b

d

P (B and C) 503=

a

c

e

Yes because orA B A BP P P54+ = =^ ^ ^h h h

No because orB C B CP P P!+^ ^ ^h h h

a b

c d

e

361

94

41

361

91

a

b

c d

e

The sample space has 9 possible outcomes. This is expected since 3 outcomes are in each event. So, there shall be 3 3 9# = total outcomes.

91

31

31


Y

Y

O

Y

W

O

Y - Y

Y - W

W - O

-

-

-

-

-

-

W

O

O

O

O O

O

Y Y

Y

YW

W W

W W

O

W

f

g

h i

j

Y

W

W

W

O

O

Y

Y

O

Y - W

Y - O

W - Y

W - O

O - Y

O - W

There are now 6 total outcomes.

31

31

0

P (A and C) 101=

b

d

No because orC D C DP P P!+^ ^ ^h h h

5. a

c

e

b

d

f

orA BP51=^ h

orB EP5023=^ h

orE FP2513=^ h

orC DP2512=^ h

orB DP52=^ h

orA FP51=^ h


Probability

31SERIES TOPIC

J 15

Answers

Using Our Knowledge:

Thinking More:

Thinking More:

31

31

31

61

61

8. 4.

5.

1.

2.

3.

a

b c

d e

f

Singles TrophyIndia Spain Greece

Doub

les T

roph

yIn

dia India,

IndiaSpain, India

Greece, India

Spai

n India, Spain

Spain, Spain

Greece, Spain

a b

c

d

e

+ is a mathematical symbol for the term intersection. For example, A B+ is a set which contains all the elements that sets A and B have in common.

, is a mathematical symbol for the term union. For example, A B, is a set which contains all the elements that are in A or in B or in both A and B.

catsP207=^ h

9 4

cats dogsP51+ =^ h

cats dogs 1P , =^ h

5 9

6 3

a b

c d

e f

g

409

87

203

a b

c d

e

1001

1252

252

12539

There are 3 stages.

a

c

b

d

e

boyP2512=^ h

vollyballP257=^ h

girl plays footballP5011=^ h

boy plays footballP245=^ h

football and girlP1611=^ h


Probability

32SERIES TOPIC

J 15

Notes

www.mathletics.com

Probability

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