Not a Venn diagram?

Finding Probability Using Sets Chapter 4.3 – Dealing With Uncertainty

A Simple Venn Diagram A’ S A Venn Diagram: a diagram in which sets are represented by geometrical shapes A’ A S

Set Notation In mathematics, curly brackets (braces) are used to denote a set of items Ex: these are sets of numbers 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 called elements.

Intersection of Sets S A ∩ B B A Given two sets, A and B, the set of common elements is called the intersection of A and B, is written as A ∩ B (“A intersect B”). S A ∩ B B A

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 Let 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? { } or Ø (the empty set, sounds like the vowel sound in bird or hurt) d) A ∩B ∩D? {10} or D

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 members of set A or members of set B (or both). So… A U B = {elements in A OR B (or both)} 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?

Disjoint Sets A and B are disjoint sets if they have no elements in common n(A ∩ B) = 0 The number of elements A ∩ B is 0 The intersection of A and B is empty A ∩ B = Ø What would the Venn diagram look like?

Disjoint Sets (continued) A Venn diagram for two disjoint sets might look like this: S B A

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) Alternatively: n(A ∩ B) = n(A) + n(B) – n(A U B) P(A ∩ B) = P(A) + P(B) – P(A U B)

Mutually Exclusive Events Mutually exclusive events have no outcomes in common A and B are mutually exclusive events if and only if (A ∩ B) = Ø e.g., flipping a head or tail e.g., drawing a red card or a black card So for mutually exclusive events A and B, n(A U B) = n(A) + n(B)

Example 3 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 n(R U F) = n(R) + n(F) – n(R ∩ F) = n(red) + n(face) – n(red face) = 26 + 12 – 6 = 32 What is the probability of picking a red card or a face card from a standard deck? P(R U F) = 32/52 = 8/13 or 0.62

Example 4 A survey of 100 students 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 A survey of 100 students How many students study English only? French only? Math only? We need to draw a Venn diagram

Example 4: what do we know? n(E ∩ M ∩ F) = 5 5

Example 4: what else do we know? 5 n(E ∩ M ∩ F) = 5 n(M ∩ E) = 30 Therefore, the number of students in E and M, but not in F is 25. 25

Example 4 (continued) n(F ∩ E) = 50 25 n(F ∩ E) = 50 Therefore, the number of students who take English and French, but not in Math is 45. 45 5 n(E) = 80

Example 4 – completed Venn Diagram 5 25 45 17 1 2

MSIP / Home Learning Read through Examples 2-3 on pp. 223-227 Complete p. 228 #1, 2, 4, 7, 8, 10–14, 17

Warm up What is the number of cards that are either even numbers (2, 4, 6, 8, 10) or clubs? What is the probability of picking such a card from a standard deck? Use n(E U C) = n(E) + n(C) – n(E ∩ C) = n(even) + n(clubs) – n(even clubs) = 20 + 13 – 5 = 28 Probability? P(E U C) = 28/52 = 7/13

Conditional Probability Chapter 4.4 – Dealing with Uncertainty Learning goal: calculate probabilities when one event is affected by the occurrence of another Questions? p. 228 #1, 2, 4, 7, 8, 10–14, 17 MSIP/Home Learning: pp. 235 – 238 #1, 2, 4, 6, 7, 9, 10, 19

Definition of Conditional Probability In some situations, knowing that one event has occurred affects the probability that another event will occur. Examples: Weather Traffic lights Star athletes’ performance Dealing cards (no replacement)

Conditional Probability Formula The probability that event B occurs given that event A has occurred is: P(B | A) = P(A ∩ B) P(A)

Example 1a Light 1 and Light 2 are both green 60% of the time. Light 1 is green 80% of the time. What is the probability that Light 2 is green given that Light 1 is green?

Example 1b The probability that it snows Saturday and Sunday is 0.2. The probability that it snows Saturday is 0.8. What is the probability that it snows Sunday given that it snowed Saturday.

Multiplication Law for Conditional Probability The probability of events A and B both occurring, when B is conditional on A is: P(A ∩ B) = P(B|A) x P(A)

Example 2 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(A ∩ B) = P(B | A) x P(A) P(1st FC ∩ 2nd FC) = P(2nd FC | 1st FC) x P(1st FC) = 11 x 12 51 52 = 132 2652 = 11 or 0.05 221

Example 3 100 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 5 100 Students surveyed Refer to yesterday’s Venn diagram. What is the probability that a student takes Mathematics given that he or she takes English?

Example 3 – Venn Diagram E 5 25 45 1 2 17

Another Example (continued) To answer the question, we need to find P(Math | English). We know... P(Math | English) = P(Math ∩ English) P(English) Therefore… P(Math | English) = 0.3 = 3 or 0.375 0.8 8

MSIP / Home Learning Read Examples 1-3, pp. 231 – 234