10A Mathematics Semester Two

Module 2

Sets, Venn Diagrams, Real Numbers and Probability

Name:

Home Group:

Contents:

• Set 1 Introduction Sets & Venn Diagrams

• Set 2 Number Sets

• Set 3 Simplifying Surds

• Set 4 Modulus

• Set 5 Probability

• Set 6 Independent Events

SAC Calculator active. (Set 1 to Set 6 inclusive)

Maths On Line Lessons

You may find these lessons helpful. 5218, 5222, 5368, 5372, 5620

A set is a collection of objects or things. Each object is called an element or member of the set.

When we record a set, we write its members within curly brackets, with commas between them. We often use a capital letter to represent a set so that we can refer to it easily. For example, suppose F is the set of all factors of 12. We write F = {1, 2, 3, 4, 6, 12}.

SET NOTATION

• E reads is an element of or is in • 0 reads is not an element of or is not in

• {} or 0 is used to represent an empty set which has no elements or members • n(S) reads the number of elements in set S

For the set F = {1, 2, 3, 4, 6, 12}, we can write 4 E F, 7 0 F, and n(F) = 6.

SUBSETS

I Set P is a subset of set Q if every element of P is also an element of Q. We write P c Q.

For example, suppose A = {3, 8} and B = {2, 3, 6, 8, 9}. Every element of A is also an element of B, so A C B.

UNION AND INTERSECTION

If P and Q are two sets then: • P n Q is the intersection of P and Q, and consists of all

elements which are in both P and Q. • PUQ is the union of P and Q, and consists of all elements

which are in either P or Q.

For example, suppose P = {1, 2, 4, 5, 6, 8} and Q = {13, 2, 3, 5, 6, 7}.

• P n Q = {2, 5, 6} as 2, 5, and 6 are in both sets • P U Q = {0, 1, 2, 3, 4, 5, 6, 7, 8} as these are the elements which are in P or Q.

UNIVERSAL SETS

Suppose we are only interested in the single digit positive whole numbers {1, 2, 3, 4, 5, 6, 7, 8, 9}. We would call this set our universal set.

I The universal set under consideration is represented by . I If we are considering the possible results of rolling an ordinary die, the universal set would be U = {1, 2, 3, 4, 5, 6}.

COMPLEMENT OF A SET

I The complement of A, denoted A', is the set of all elements of that are not in A. I If = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 3, 5, 7, 9} then the complementary set of A is A' = {2, 4, 6, 8}.

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A Venn diagram consists of a universal set represented by a rectangle, and sets within it that are generally represented by circles.

The Venn diagram alongside shows set A within the universal set

The complement of A is the shaded region outside the circle which represents A.

Suppose = {1, 3, 4, 6, 91, A = {1, 6, 91, and A' = {3, 41.

These sets are represented in the Venn diagram alongside.

If B is a subset of A, then every element of B is also in A.

The circle representing B is placed within the circle representing A.

A A'

9 4

leo A

The intersection of two sets A and B consists of all elements common to both A and B.

A n B is the shaded region where the circles representing A and B overlap.

The union of two sets A and B consists of all elements in A or B or both.

A U B is the shaded region which includes everywhere in either circle.

Disjoint or mutually exclusive sets do not have common elements.

They are represented by non-overlapping circles.

If the sets are disjoint and exhaustive then B = A' and A U B =

We can represent this situation without using circles as shown.

A B

Example:

A = {5,10,15}

B = {2,4,6,8,10,14}

C= {3,6,9,12,15}

= {1,2,3,4, 14,15}

1) Represent the above sets on a Venn diagram.

2) find:

a) Au B

b) Au C

c) A n B d) A nC

e) B nC

3) State True or False

a) {4,6,81 B

b) Cc

c) B A

4) B' means the Complement of B ie all elements outside of B Now find:

a) B'

b) (AuBY

5) n(B) means the number of elements contained within set B Now find

a) n(B)

b) n(A')

c)

6) The empty set or Null set is represented using

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Real Numbers

Types of Real Numbers

1. Natural numbers (N): 1, 2, 3, 4, 5, ...

2. Integer numbers (Z): 0, ±1, ±2, ±3, ±4, ±5,

REMARK: Any natural number is an integer number, but not any integer number is a natural number.

3. Rational numbers (Q):

In r = —Th , where m E Z, n E N

EXAMPLES:

1 7 11 2 2 1 —2 5 - 2 = 1. 0.2 = 37) = 0.222 . . . = 0.2 = 5 0.999... = 0. = 1

REMARK: Any integer number is a rational number, but not any rational number is an integer.

4. Irrational Numbers. These are numbers that cannot be expressed as a ratio of integers.

EXAMPLES:

- 13/ 7F \Fr 2'/ 2

The set of all real numbers is denoted by R. The real numbers can be represented by points on a line which is called a coordinate line, or a real number line, or simply a real line:

111 —4.9 —4.7 —3.1725 -1 8 4 I

2 fV5 4.2 4.4 4.9999

—2.63 IT II I I tilt I 1 1 1 11 1 11 1 1 1 III

—5\ —4 —3 —2 —1 Ot 1 2 3 47 5 —4.85 0.3 4.3 4.5

Every real number has a decimal representation. If the number is rational, then its correspond-ing decimal is repeating. For example,

1 2 = 0.500... = 0.5171= 0.4, = 0.66666... = 0.

157 = 0.3171717... = 0.317,

9 = 1.285714285714... = 1.285714

495 7 If the number is irrational, the decimal representation is nonrepeating:

= 1.414213562373095... = 3.141592653589793...

Following is a list of some special number sets you should be familiar with:

N = {1, 2, 3, 4, 5, 6, 7, ....}

is the set of all natural or counting numbers.

Z = {0, ±1, ±2, ±3, ±4, ....}

is the set of all integers.

Z+ = {1, 2, 3, 4, 5, 6, 7, ....}

is the set of all positive integers.

Z- = { —1, —2, —3, —4, —5, ....}

is the set of all negative integers.

Q is the set of all rational numbers, which are numbers that can be written in the form where p and q are integers and q 0 0.

Q' is called the set of all irrational numbers. Numbers like \,/, .0, and ir belong to Q'.

R is the set of all real numbers, which are the numbers that can be placed on the number line.

All of these special sets have infinitely many elements. We call them infinite sets, and say that n(S) = oo.

Real Number Sets

The Real Number System contains the set of rational and irrational numbers. It is denoted by the symbol R. The set of real numbers contains a number of subsets which can be classified as shown in the chart below.

Real numbers R

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The MODULUS of a number defines the magnitude of that number and is always defined positive. Modulus measures the magnitude (distance) of the number from the origin (le 0 on the real number line).

13 1 defines the modulus of 3 ; 1 3 1 = 3

1 — 4 1 defines the modulus of —4 ; 1 — 4 1 = 4

1

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Probability The study of probability deals with the chance or likelihood of an event happening. For eve' event we can assign a number which lies between 0 and 1 inclusive.

An impossible event has 0% chance of happening, and is assigned the probability O.

A certain event has 100% chance of happening, and is assigned the probability 1.

All other events between these two extremes can be assigned a probability between 0 and 1.

The number line below shows how we could interpret different probabilities:

not likely

likely to happen to happen

0 r--

impossible certain

very unlikely equal chance of

very likely to happen

happening as not happening to happen

The sample space of an experiment is the set of its possible outcomes. it can be represented by:

• listing the possible outcomes • a 2-dimensional grid • a Venn diagram • a two-way table • a tree diagram.

If a sample space has n outcomes which are equally likely to occur when the experiment is

performed once, then each outcome has probability —1 of occurring.

An event occurs when we obtain an outcome with a particular property or feature.

When the outcomes of an experiment are equally likely, the probability of an event E occurring is given by:

number of outcomes corresponding to E P (E) —

number of outcomes in the sample space

Suppose three coins are tossed simultaneously

a List the possible outcomes.

b Find the probability of getting: i three heads at least one head.

a The possible outcomes are: {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}.

— three heads only occurs in the outcome HHH b i P(three heads) —

8 — 8 possible outcomes

7 — all outcomes except TTT contain at least one head P(at least one head) = —

8 — 8 possible outcomes

A die has the numbers 0, 0, 1, 1, 4, and 5. It is rolled twice. a Illustrate the possible outcomes using a 2-dimensional grid. b Hence find the probability of getting:

i a total of 5 tvvo numbers which are the same.

a roll 2

b There are 6 x 6 = 36 possible outcomes. P(total of 5) - 8 {those with a x 1 - 36 - 2 - 9

P(same numbers) _ {those circled} - 36 - 5 - 18

roll 1 0 0 1 1 4 5

In a library group of 50 readers, 31 like science fiction, 20 like detective stories, and 12 dislike both.

a Draw a Venn diagram to represent this information. b If a reader is randomly selected, find the probability that he or she:

i likes science fiction and detective stories ii likes exactly one of science fiction and detective stories.

a Let S represent readers who like science fiction, and D represent readers who like detective stories. We are given that a + b = 31

b + c = 20 a + b + c = 50 — 12 = 38

c = 38 — 31 = 7 b = 13 and a = 18

i P(Iikes both) = 18 + 7 P(Iikes exactly one) = —

50 1 = -2

S (18) OD (12)

1 a List all possible orderings of the letters 0, D, and G.

b The three letters 0, D, and G are placed at random in a row. Find the probability of i spelling DOG ii 0 appearing first

iii 0 not appearing first iv spelling DOG or GOD.

2 a List the 8 possible 3-child families, according to the gender of the children. For example, BGB means "the first is a boy, the second is a girl, and the third is a boy".

b Assuming that each of these outcomes is equally likely to occur, determine the probability that a randomly selected 3-child family consists of:

i all boys ii all girls iii boy, then girl, then girl iv two girls and a boy v a girl for the eldest vi at least one boy.

3 a List the 24 different orders in which Alex, Bodi, Claire, and Daniel may sit in a row.

b If the four friends sit randomly in a row, determine the probability that: i Alex is on one of the ends ii Claire and Daniel are on the ends

Bodi is on an end, and Claire is seated next to him iv Alex and Claire are next to each other.

4 a Draw the grid of the sample space when a 10-cent and a 50-cent coin are tossed simultaneously.

b Hence, determine the probability of getting:

i two heads ii two tails iii exactly one head iv at least one head.

5 A coin and a pentagonal spinner with sectors 1, 2, 3, 4, and 5 are tossed and spun respectively

a Draw a grid to illustrate the sample space of possible outcomes.

b How many outcomes are possible?

c Use your grid to determine the chance of getting: i a head and a 4 a tail and an odd number

iii an even number iv a tail or a 3.

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6 A pair of dice is rolled.

a Display the possible outcomes on a grid.

b Hence, determine the probability of:

i one die showing a 4 and the other a 5 ii both dice showing the same number

iii at least one of the dice showing a 3 v both dice showing even numbers

_ — • 1 • • ,` • • • iv either a 4 or 6 being displayed vi the sum of the numbers being 7.

Set 5 Probability

7 A die has the numbers 1, 2, 2, 2, 5, and 6. It is rolled twice.

a Draw a grid to illustrate the possible outcomes.

b Hence, find the probability of getting: i a 2 and a 5 ii a total of 7

iii numbers which are the same iv at least one 6 v at least one prime number vi numbers whose product is at least 5.

8 The Venn diagram shows the sports played by the students in a Year 10 class.

a How many students are in the class?

b A student is chosen at random. Find the probability that the student:

i plays football

iii plays football or rugby

FSDR (9)

F a football R E rugby

ii plays both codes

iv plays exactly one of these sports.

9 In a class of 24 students, 10 take Biology, 12 take Chemistry, and 5 take neither Biology nor Chemistry.

a Copy and complete the Venn diagram. b Find the probability that a student picked at

random from the class takes: • Lnenustry, out not Diology

ii Chemistry and Biology.

1 0 50 tourists went on a 'thrill seekers' holiday. 40 went white-water rafting, 21 went paragliding, and each tourist did at least one of these activities. Find the probability that a randomly selected tourist:

a participated in both activities b went white-water rafting but not oaragliding.

1 a {ODG, OGD, DOG, DGO, GOD, GDO}

b I II A III Iv A 2 a {BBB, GBB, BGB, BBG, BGG, GBG, GOB, GGG}

b I A II A III A lv v A vl

3 a {ABCD, ABDC, ACBD, ACDB, ADBC, ADCB, BACD, BADC, BCAD, BCDA, BDAC, BDCA, CABD, CADB, CBAD, CBDA, CDAB, CDBA, DABC, DACB, DBAC, DBCA, DCAB, DCBA}

b I A II I III iv

6 6

4 a 50c coin b I II ! 4 4

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7 a die 2

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Answers

INDEPENDENT EVENTS

Two events are independent if the occurrence of each event does not affect the occurrence of the other.

For example, suppose the spinners alongside are spun simultaneously

Let A be the event the first spinner lands on red, and B be the event the second spinner lands on an even number.

Whether A occurs or not does not affect the probability of B occurring. Likewise, whether B occurs or not does not affect the probability of A occurring. Therefore, A and B are independent events.

Now, we know that P(A) = and P(B) = t. By listing each of the possible outcomes, we can see that P(A and B) =

{R1, R2, R3, R4, R5, Gl, G2, G3, G4, G5, Bl, B2, B3, B4, B5}

We notice that = x t, so P(A and B) = P(A) x P(B).

If two events A and B are independent, then P(A and B) = P(A) x P(B).

A coin is tossed and a die rolled simultaneously. Find the probability that a tail and a '2' result.

'Getting a tail' and 'rolling a 2' are independent events.

P(a tail and a `2') = P(a tail) x P(a '2')

This rule can be extended to any number of independent events.

For example:

If A, B, and C are all independent events, then

P(A and B and C) = P(A) x P(B) x P(C).

A marble is selected at random from each of the bags alongside.

a Draw a tree diagram to display the possible outcomes.

b Find the probability of obtaining a green marble and a yellow marble.

• *

Bag A Bag B

Bag A Bag B

4

G < -7 Y 2

< 5 5 4

■ Each branch of the

tree diagram represents a

possible outcome.

,4

If 2 or more branches satisfy the event, the

probabilities are added. ■ P(a green and a yellow)

2

= P(GY or YG) {branches marked .7} _3 ± 2 x 4 — 5 7 5 7

11 — 35

a

b

1 A die is rolled, and the spinner alongside is spun. Find the probability of obtaining:

a a '5' and a green

b an even number and a non-red.

2 A disc is selected from each of the containers alongside. Find the probability of selecting:

a a red disc from A and a green disc from B

b a blue disc from both containers.

Set 6 Independent Events

3 A school has two photocopiers. On any one day, machine A ' '-apk.....'""11.1 has an 8% chance of having a paper jam, and machine B has ,

tv.-1 ' a 12% chance of having a paper jam. Determine the probability that, on any one day, both L — machines will:

a have paper jams b work uninterrupted.

4 A boy and a girl were each asked what day of the week they were born. Find the probability that:

a the boy was born on a Monday and the girl was born on a Wednesday

b the boy was born on a weekend, but the girl was not

c both children were born on a weekday.

8 A ticket is selected at random from each of the boxes alongside.

a Draw a tree diagram to display the possible outcomes. 1111100"0

11011-,001111:1.464 It 1:1-61 b Find the probability of obtaining: Box A Box B

i a blue ticket and a green ticket ii two tickets of the same colour

III exactly one pink ticket.

c Find the sum of the probabilities in b . Explain your result.

9 Sharon, Christine, and Keith have agreed to meet at a restaurant for lunch. Sharon has probability 0.8 of being on time, Christine has probability 0.7 of being on time, and Keith has probability 0.4 of being on time.

a Draw a tree diagram to display the possible outcomes.

b Determine the probability that at least 2 people will arrive on time.

a b 2 a 2 48 16 9 b

8 a Box A Box B b I 2 Il

9 63

2 .......4• P iii 29 ....., 9 3

" . 63 P — --.. B 4 z ---. 4 2 5 ---..... G

....., 9 2_,...-+•3 P

4 —1.- B..,,, 5-.- B < 5"-"sn G

4 N 2 .......■ P

,..., 9 3 G„..„=1 5-••• B

5 ----... G

C 1; this is because every possible outcome is covered by these three events.

3 a = 0 0096 b 506 — 0.8096 625 • 625

4 1 b 10 c 25 49 49 49

9 a Sharon Christine Keith b 0.712

On fime ..< °A .4- On time

....• 0.7 0.6,- Late

../., On time < 0.3,, .0 4 -.- On time

0.8 Late {

<

0.6,, Late

0.2 0 4 -.. On time On time .< n'g

N 0.7'r ,,,,,,, Late Late <

0.3,, .0 4 -■ Go time Late {

0.6,, Late

Answers