Warm up The probability of event A is given by

P(A) = n(A) = 8 = 2

n(S) 52 13 What could event A be? What is the Sample Space, S?

AGENDA

Play the Coffee Game on p. 203 (15 mins) Answer Discussion Questions on p. 204 (5

mins) Take up Discussion Questions (5 mins) Introduction to Set Theory

Coffee Game

Record your results in a table similar to:

Wk 1 2 3 4 5 6 7 8 9 10

Mon X Y

Tue Y Y

Wed X Y

Thu X X

Fri Y X

MS Excel Formulas to generate random integers Random 0 or 1 (coin toss, predict gender of a baby)

= ROUND ( RAND ( ) * ( 1 ) , 0 )

Random integer between 1 and 5 (football kicker p. 211 #9)

= ROUND ( RAND ( ) * ( 4 ) + 1 , 0 )

Random integer between 1 and 6 (die)

= ROUND ( RAND ( ) * ( 5 ) + 1 , 0 )

Random integer between 1 and n

= ROUND ( RAND ( ) * ( n-1 ) + 1 , 0 )

Type a formula into a cell, then copy and paste to a group of cells to simulate multiple trials (6, 10, 20, 100, etc.)

e.g., 4.1 random numbers.xls

Finding Probability Using Sets

Chapter 4.3 – Dealing With Uncertainty

Mathematics of Data Management (Nelson)

MDM 4U

John Venn 1834 -1923 “Of spare build, he

was throughout his life a fine walker and mountain climber, a keen botanist, and an excellent talker and linguist”-- John Archibald Venn (John Venn’s son), writing about his father

A Simple Venn Diagram Venn Diagram: a diagram in which sets are

represented by shaded or coloured geometrical shapes.

A’

A

S

Set Notation

In mathematics, curly brackets are used to denote a set of items

e.g., A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} B = {2, 4, 6, 8, 10} C = {1, 2, 3, 4, 5} D = {10} The items in a set are commonly called

elements.

Intersection of Sets Given two sets, A and B, the set of common

elements is called the intersection of A and B, is written as A ∩ B.

SA ∩ B

Intersection of Sets (continued) Elements that belong to the set A ∩ B are

members of set A and members of set B.

So… A ∩ B = {elements in both A AND B}

S A ∩ B

Example 1 - Intersection

Recall

A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} C = {1, 2, 3, 4, 5}

B = {2, 4, 6, 8, 10} D = {10} a) What is A ∩ B? {2, 4, 6, 8, 10} or B b) B ∩ C? {2, 4} c) C ∩ D? Ø (the empty set, ĭ) d) A ∩B ∩D? {10}

Union of Sets The set formed by combining the elements of A

with those in B is called the union of A and B, and is written A U B.

SA U B

Union of Sets (continued) Elements that belong to the set A U B are either

members of set A or members of set B (or both).

So… A U B = {elements in A OR B}

SA U B

Example 2 - Union

A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} B = {2, 4, 6, 8, 10}

C = {1, 2, 3, 4, 5} D = {10} a) What is A U B? {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} or A b) B U C? {1, 2, 3, 4, 5, 6, 8, 10} c) C U D? {1, 2, 3, 4, 5, 10} d) B U C U D? {1, 2, 3, 4, 5, 6, 8, 10}

Disjoint Sets

If set A and set B have no elements in common (that is, if n(A ∩ B) = 0), then A and B are said to be disjoint sets and their intersection is the empty set, Ø.

Another way of writing this: A ∩ B = Ø

What is the symbol Ø ?

http://encyclopedia.thefreedictionary.com/%D8

Disjoint Sets (continued) A Venn diagram for two disjoint sets might look

like this:

S

BA

The Additive Principle

Remember: n(A) is the number of elements in set A P(A) is the probability of event A

The Additive Principle for the Union of Two Sets: n(A U B) = n(A) + n(B) – n(A ∩ B) P(A U B) = P(A) + P(B) – P(A ∩ B)

Mutually Exclusive Events

A and B are mutually exclusive events if and only if: (A ∩ B) = Ø(i.e., they have no elements in common)

This means that for mutually exclusive events A and B, n(A U B) = n(A) + n(B)

Example 1 What is the number of cards that are either red

cards or face cards? Let R be the set of red cards, F the set of face cards If we have “or” we are looking at union

n(R U F) = n(R) + n(F) – n(R ∩ F) = n(red) + n(face) – n(red face) = 26 + 12 – 6 = 32

Probability? P(R U F) = 32/52 = 8/13

Example 2 A survey of 100

students How many

students are enrolled in English and no other course?

Course Taken No. of students

English 80

Mathematics 33

French 68

English and Mathematics

30

French and Mathematics

6

English and French

50

All three courses 5

How many only study French?

Example 2: what do we know? n(E ∩ M ∩ F) = 5

M

F

E

5

Example 2: what else do we know? n(E ∩ M ∩ F) = 5

M

F

E

5

n(M ∩ E) = 30

Therefore, the number of students in E and M, but not in F is 25.

25

Example 2 (continued)

n(F ∩ E) = 50

Therefore, the number of students who take English and French, but not in Math is 45.

M

F

E

5

25

45

n(E) = 80

5

Example 2 – completed Venn Diagram

M

F

E

5

25

45

5

1

17

2

MSIP / Homework

Read through Examples 1-3 on pp. 223-227 (in some ways, Example 1 is very similar to the example we have just seen).

Exercises: p. 228 #1, 2, 4, 8, 9, 10, 11, 14, 17 Quiz Thurs. on 4.1 – 4.4

Conditional Probability

Chapter 4.4 – Dealing with UncertaintyMathematics of Data Management (Nelson)MDM 4U

Definition of Conditional Probability The conditional probability of event B, given

that event A has occurred, is given by:

P(B | A) = P(A ∩ B) P(A)

Therefore, conditional probability deals with determining the probability of an event given that another event has already happened.

Multiplication Law for Conditional Probability

The probability of events A and B occurring, given that A has occurred, is given by

P(A ∩ B) = P(B|A) x P(A)

Example a) What is the probability of drawing 2 face cards in a row from

a deck of 52 playing cards if the first card is not replaced?

P(1st FC ∩ 2nd FC) = P(2nd FC | 1st FC) x P(1st FC)

= 11 x 12 51 52

= 132 2652

= 11 221

Another Example 100 Students surveyed

Course Taken No. of students

English 80

Mathematics 33

French 68

English and Mathematics

30

French and Mathematics

6

English and French

50

All three courses

5

b) What is the probability that a student takes Mathematics given that he or she also takes English?

a) Draw a Venn Diagram that represents this situation.

Another Example – Venn Diagram

M

F

E

5

25

45

5

1

2

17

Another Example (continued)

To answer the question in (b), we need to find P(Math|English).

We know... P(Math|English) = P(Math ∩ English)

P(English) Therefore…

P(Math|English) = 30 / 100 = 30 x 100 = 3 80 / 100 100 80 8

MSIP / Homework

Read Examples 1-3, pp. 231 – 234 p. 235 – 238 #2, 4, 6-10, 14, 19