Finding Probability Using Sets

1
Finding Probability Using Sets

• Chapter 4.3 Dealing With Uncertainty
• Mathematics of Data Management (Nelson)
• MDM 4U
• Authors Gary Greer and James Gauthier
• (with K. Myers)

2
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 Venns son),
  writing about his father

3
A Simple Venn Diagram

• Venn Diagram a diagram in which sets are
  represented by shaded or coloured geometrical
  shapes.

4
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 n B.

5
Intersection of Sets (continued)

• Elements that belong to the set A n B are members
  of set A and members of set B.
• So A n B elements in both A AND B

6
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.

7
Union of Sets (continued)

• Elements that belong to the set A U B are either
  members of set A or members of set B.
• So A U B elements in A OR B

8
Disjoint Sets

• If set A and set B have no elements in common
  (that is, if n(A n 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 n B Ø
• What is the symbol Ø ?
• http//encyclopedia.thefreedictionary.com/D8

9
Disjoint Sets (continued)

• A Venn diagram for two disjoint sets might look
  like this

10
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 n B)
• P(A U B) P(A) P(B) P(A n B)
• http//stat-www.berkeley.edu/stark
  /Java/Venn.htm

11
Mutually Exclusive Events

• A and B are mutually exclusive events if and only
  if (A n B) Ø
• (i.e., they have no elements in common)
• This means that for mutually exclusive events n(A
  U B) n(A) n(B)

12
Example

• What is the number of cards that are either red
  cards or face cards?
• If we have or we are looking at union
• n(A U B) n(A) n(B) n(A n B)
• n(A U B) n(red) n(face) n(both)
• n(A U B) 12 26 6
• n(A U B) 32
• Probability?
• p(A U B) 32/52 8/13

13
Example

• A survey of 100 students

• How many students are enrolled in English and no
  other course?

• How many only study French?

14
Example what do we know?

• n(E n M n F) 5

5
15
Example what else do we know?

• n(E n M n F) 5

• n(M n E) 30

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

25
16
Example (continued)

• n(F n E) 50

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

45
5

• n(E) 80

17
Example completed Venn Diagram
17
1
2
18
Exercises and Homework

• 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, 3, 4
• Exercises p. 229, 6, 7, 9, 10, 12

19
Conditional Probability

• Chapter 4.4 Dealing with Uncertainty
• Mathematics of Data Management (Nelson)
• MDM 4U
• Author Gary Greer and James Gauthier
• (with K. Myers)

20
Definition of Conditional Probability

• The conditional probability of event B, given
  that event A has occurred, is given by
• P(BA) P(A n B)
• P(A)
• Therefore, conditional probability deals with
  determining the probability of an event given
  that another event has already happened.

21
Multiplication Law for Conditional Probability

• The probability of events A and B occurring,
  given that A has occurred, is given by
• P(A n B) P(BA) x P(A)

22
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 n 2nd FC) P(2nd FC 1st FC) x
  P(1st FC)
• 11 x 12
• 51
  52
• 132
• 2652
• 11
• 221

23
Another Example

• 100 Students surveyed

• a) Draw a Venn Diagram that represents this
  situation.

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

24
Another Example Venn Diagram
25
Another Example (continued)

• To answer the question in (b), we need to find
  P(MathEnglish).
• We know...
• P(MathEnglish) P(Math n English)
• P(English)
• Therefore
• P(MathEnglish) 30 / 100 30 x 100 3
• 80 / 100 100 80 8

26
Exercises / Homework

• Homework
• Read Examples 1-3, pp. 231 234
• Exercises p. 235, 1, 2, 4, 5, 7
• Exercises p. 236, 9
• Exercises p. 237, 13
• (we might do this one together)