1. Not a Venn diagram?

2. Finding Probability Using Sets
Chapter 4.3 – Dealing With Uncertainty

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

4. 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.

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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?

11. 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?

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

13. 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)

14. 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)

15. 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)
= – 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

16. 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

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

18. 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

19. Example 4 (continued) n(F ∩ E) = 5025
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

20. Example 4 – completed Venn Diagram5 25 45 17 1 2

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

23. 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)
= – 5
= 28
Probability?
P(E U C) = 28/52 = 7/13

24. 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

25. 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)

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

27. 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?

28. 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.

29. 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)

30. 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
= 132
2652
= or 0.05
221

31. 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?

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

33. 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 = or 0.375

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