Warm up The probability of event A is given by P(A) = n(A) = 8 = 2 n(S) 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 MonXY TueYY WedXY ThuXX FriYX

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.xls4.1 random numbers.xls

Finding Probability Using Sets Chapter 4.3 – Dealing With Uncertainty Mathematics of Data Management (Nelson) MDM 4U

John Venn “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. S A ∩ 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. S A 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} S A 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 ?

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)  = – 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 TakenNo. 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

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 n(E) = 80 5

Example 2 – completed Venn Diagram M F E

MSIP / Homework Read through Examples 1-3 on pp (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 Uncertainty Mathematics 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(1 st FC ∩ 2 nd FC) = P(2 nd FC | 1 st FC) x P(1 st FC) = 11 x = =

Another Example 100 Students surveyed Course TakenNo. 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

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 /

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