Not a Venn diagram?

Finding Probability Using Sets

Chapter 4.3 – Dealing With Uncertainty

A Simple Venn Diagram 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 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”).

SA ∩ B

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

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.

SA 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)}

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

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

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)

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 How many

students study English only? French only? Math only?

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

We need to draw a Venn diagram

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

M

F

E

5

Example 4: 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 4 (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 4 – completed Venn Diagram

M

F

E

5

25

45

5

1

17

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 UncertaintyLearning goal: calculate probabilities when one event is affected by the occurrence of anotherQuestions? p. 228 #1, 2, 4, 7, 8, 10–14, 17MSIP/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

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

M

F

E

5

25

45

5

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 pp. 235 – 238 #1, 2, 4, 6, 7, 9, 10, 19