Sets and Probability

1
Lesson 8

• Sets and Probability

2
Set and Set Operations

• A set is any well-defined collection of objects
  the objects are called the elements or members of
  the set. A set is usually denoted by a Capital
  letter such as A, Z, Y.., whereas lower-case
  letters a, z, y are usually used to denote
  elements of sets.
• Two main ways to specify a set.
• List its elements Aa ,z ,ymeans A is the set
  whose elements are the letters a, z, and y.

3
Continued

• 2. State the properties which characterize the
  elements in the set. Example
• Byy is an odd integer, ygt0
• Which reads
• B is the set of y such that y is an odd integer
  and y gt 0)
• The colon is read as such that and the comma as
  and.

4
Continued

• Two sets Y and Z are equal, written as YZ if
  they both have the same elements. The negation of
  YZ is written as Y?Z.
• If x belongs to Y, that is x is an element of Y
  is written as
• x ?Y and x, z ? Y means x and z belong to Y
• x ?Y means x doesnt belong to Y

5
Continued

• The universal set is the set containing
  everything in a given context. We denote the
  universal set by U.
• The empty set (or Null set) is the set containing
  no elements. It is denoted by Ø.
• SUBSETS
• If every element in set Y is also an element in
  set X then Y is called a subset of X
• Y ? X means Y is contained by X
• X ? Y means X contains Y

6
Example

• A1,3,5,7 B1,2,3,5 C1,5
• Then C ? A or A ? C
• That means C is contained by A or A contains C.
• Also
• C ? B or B ? C.
• The Universal set U1,2,3,5,7
• PROPER SUBSETS
• If set Y is a subset of X and there is at least 1
  element in X that is no in Y, than we say that Y
  is a proper subset of X denoted Y? X

7
Continued

• Two sets X and Y are disjoint if they have no
  elements in common
• The complement of set A is the set containing all
  elements in the universal set U that are not
  members of A. Denoted A (or see next slide )

Y
X
_
8
Complement of an Event (2)

• The complement of event A is defined to be the
  event consisting of all sample points that are
  not in A.
• The complement of A is denoted by Ac.
• The Venn diagram below illustrates the concept of
  a complement.

Sample Space S
Event A
Ac
9
Venn Diagrams and Probability
U the universal set
10
Venn Diagrams and Probability
U
Set A
A
11
Venn Diagrams and Probability
U
A
A
12
Venn Diagrams and Probability
U all students in class
A
A Female
A Male
A
13
Intersection

• The intersection of two sets, A and B, is the set
  containing all elements that are members of both
  A and B. Denoted by A I B
• That is A ? B x x ? A and x ? B

A
RED intersection of A and B
B
14
Union

• The union of two sets, A and B, is the set
  containing all elements that are members of
  either A or B. Denoted by A U B
• That is A ? B x x ? A or x ?B. Or means
  and/or in this case.

A
Union total patterned area (red and blue)
B
15
Examples

• If U is all students in class and M stands for
  male and F stands for female Then M ? F U.
• Also M ? F ? since it is not possible to belong
  to both M and F.
• If the sets corresponding to events X and Y are
  disjoint, that is X ? Y ?, we say that that the
  events are mutually exclusive.

16
Properties of Set Operations

• Le U be a universal set. If X,Y and Z are subsets
  of U then
• Commutative Property for Union of Sets
• X ? Y Y ? X
• Commutative Property for Intersections of Sets
• X ? Y Y ? X
• Associative Property for Union Sets
• (X ?Y) ? ZX ?(Y ? Z)
• Associative Property for Intersections of Sets
• (X ? Y) ? ZX ? (Y ? Z)

17
Continued

• Distributive Properties
• X ? (Y ? Z)(X ?Y) ? (X ? Z)
• X ? (Y ? Z)(X ? Y) ? (X ? Z)
• DeMorgans laws

18
Example

• DeMorgans laws

S
5
A
B
3
1
2
19
Examples

• A1,2 B2,3 S1,2,3,5
• DeMorgans laws

20
PROBABILITY

• Probability also is an aid in decision-making
  under conditions of uncertainty
• Probability is a numerical measure of the
  likelihood that an event will occur.
• Probability values are always assigned on a scale
  from 0 to 1.
• A probability near 0 indicates an event is very
  unlikely to occur.
• A probability near 1 indicates an event is almost
  certain to occur.
• A probability of 0.5 indicates the occurrence of
  the event is just as likely as it is unlikely.

21
Continued

• Probability provides the foundations for
  statistical inference(3411) (hypothesis testing
  and interval estimation)
• The purpose of statistical inference is to obtain
  information about a population from information
  contained in a sample
• A population is the set of all the elements of
  interest.
• A sample is a subset of the population.

22
An Experiment

• An experiment is any process that generates
  well-defined outcomes. (Ex Toss a coin, Roll a
  die, Select a part for inspection)
• A procedure that produces an outcome
• Not perfectly predictable in advance
• Head-Tail- 1,2,3,4,5,6 Defective no defective
• All experimental outcomes are predictable but any
  given outcome cannot be predicted with certainty.

23
More Definitions

• We define an experiment as a process that leads
  to one of several possible outcomes. An outcome
  of an experiment is some observation or
  measurement.
• The sample space is the universal set, X,
  pertinent to a given experiment. The sample
  space is the set of all possible outcomes of an
  experiment.
• A sample point is an element of the sample space,
  any one particular experimental outcome.

24
Random Experiment

• An event is a subset of a sample space. It is a
  set of basic outcomes.
• Happens or not, each time random experiment is
  run
• Formally a collection of outcomes from sample
  space
• A yes or no situation if the outcome is in the
  list, the event happens
• Each random experiment has many different events
  of interest
• Example tossing a coin - the event Head
• Probability of an Event
• A number between 0 ( NEVER HAPPENS ) and
• 1 ( ALWAYS HAPPENS )
• The likelihood of occurrence of an event

25
A Counting Rule for Multiple-Step Experiments

• If an experiment consists of a sequence of k
  steps in which there are n1 possible results for
  the first step, n2 possible results for the
  second step, and so on, then the total number of
  experimental outcomes is given by (n1)(n2) . . .
  (nk).
• A helpful graphical representation of a
  multiple-step experiment is a tree diagram.

26
Example Bradley Investments

• Bradley has invested in two stocks, Markley Oil
  and
• Collins Mining. Bradley has determined that the
• possible outcomes of these investments three
  months
• from now are as follows.
• Investment Gain or Loss
• in 3 Months (in 000)
• Markley Oil Collins Mining
• 10 8
• 5 -2
• 0
• -20

27
Example Bradley Investments

• A Counting Rule for Multiple-Step Experiments
• Bradley Investments can be viewed as a two-step
  experiment it involves two stocks, each with a
  set of experimental outcomes.
• Markley Oil n1 4
• Collins Mining n2 2
• Total Number of
• Experimental Outcomes n1n2 (4)(2) 8

28
Example Bradley Investments

• Tree Diagram
• Markley Oil Collins Mining
  Experimental
• (Stage 1) (Stage 2)
  Outcomes

Gain 8
(10, 8) Gain 18,000 (10, -2) Gain
8,000 (5, 8) Gain 13,000 (5, -2)
Gain 3,000 (0, 8) Gain 8,000 (0,
-2) Lose 2,000 (-20, 8) Lose
12,000 (-20, -2) Lose 22,000
Lose 2
Gain 10
Gain 8
Lose 2
Gain 5
Gain 8
Even
Lose 2
Lose 20
Gain 8
Lose 2
29
Counting Rule for Combinations

• Another useful counting rule enables us to count
  the
• number of experimental outcomes when n objects
  are to
• be selected from a set of N objects.(Order not
  important)
• Number of combinations of N objects taken n at a
  time
• where N! N(N - 1)(N - 2) . . . (2)(1)
• n! n(n - 1)( n - 2) . . . (2)(1)
• 0! 1

30

• The odds of winning the lottery in Florida are

31
Counting Rule for Permutations

• A third useful counting rule enables us to count
  the
• number of experimental outcomes when n objects
  are to
• be selected from a set of N objects where the
  order of
• selection is important.
• Number of permutations of N objects taken n at a
  time

32
Combinations Vs Permutations

• List all combinations and all permutations of the
  4 letters A,B,C, and D When they are taken 3 at
  a time
• Combinations ABC ABD ACD BCD 4 Combinations
• Permutations ABC ABD ACD BCD
• ACB ADB ADC BDC
• BAC BAD CAD CBD
• BCA BDA CDA CDB
• CAB DAB DAC DBC
• CBA DBA DCA DCB
  24 Permutations

33
Types of Probability

• Classical Method
• Assigning probabilities based on the assumption
  of equally likely outcomes.
• Relative Frequency Method
• Assigning probabilities based on experimentation
  or historical data.
• Subjective Method
• Assigning probabilities based on the assignors
  judgment.

34
Classical Method

• If an experiment has n possible outcomes, this
  method
• would assign a probability of 1/n to each
  outcome.
• Example
• Experiment Rolling a die
• Sample Space S 1, 2, 3, 4, 5, 6
• Probabilities Each sample point has a 1/6
  chance
• of occurring.

35
Example Lucas Tool Rental

• Relative Frequency Method
• Lucas would like to assign probabilities to the
• number of floor polishers it rents per day.
  Office
• records show the following frequencies of daily
  rentals
• for the last 40 days.
• Number of Number
• Polishers Rented of Days
• 0 4
• 1 6
• 2 18
• 3 10
• 4 2

36
Example Lucas Tool Rental

• Relative Frequency Method
• The probability assignments are given by
  dividing
• the number-of-days frequencies by the total
  frequency
• (total number of days).
• Number of Number
• Polishers Rented of Days Probability
• 0 4 .10 4/40
• 1 6 .15 6/40
• 2 18 .45 etc.
• 3 10 .25
• 4 2 .05
• 40 1.00

37
Subjective Method

• When economic conditions and a companys
  circumstances change rapidly it might be
  inappropriate to assign probabilities based
  solely on historical data.
• We can use any data available as well as our
  experience and intuition, but ultimately a
  probability value should express our degree of
  belief that the experimental outcome will occur.
• The best probability estimates often are obtained
  by combining the estimates from the classical or
  relative frequency approach with the subjective
  estimates.

38
Example Bradley Investments

• Applying the subjective method, an analyst
• made the following probability assignments.
• Exper. Outcome Net Gain/Loss
  Probability
• ( 10, 8) 18,000 Gain
  .20
• ( 10, -2) 8,000 Gain
  .08
• ( 5, 8) 13,000 Gain
  .16
• ( 5, -2) 3,000 Gain
  .26
• ( 0, 8) 8,000 Gain
  .10
• ( 0, -2) 2,000 Loss
  .12
• (-20, 8) 12,000 Loss
  .02
• (-20, -2) 22,000 Loss
  .06

39
Probability of an Event

• Event any collections of possible outcomes for
  the experiment
• Is the order important ?
• Can events occur simultaneously ?
• Can the outcome of one event influence the
  likelihood on another event ?

40
Probability of Event A

• Assuming equal likelihood of the elements in the
  sample space, the probability of event A is the
  relative size of set A with respect to the size
  of the sample space, X.
• Suppose our sample space is all 100 students in a
  class, the probability of selecting one student
  with age of 21 years depends on how many 21
  year-olds are in class. Suppose 25 students are
  21 years old. The probability is 25/100 0.25.

41
Probability of Event A

• P(A) n(A)/n(S), where
• n(A) the number of elements in the set A
• n(S) the number of elements in the sample space

42
Basic Rules of Probability

• The probability of any event, such as A, always
  must satisfy
• 0 ? P(A) ? 1

0
1
impossible
certain
43
Computing Probabilities Using the Additional Rule

• Addition Rule for Mutually Exclusive Events
• General Addition Rule
• Odds

44
Union Rule for Mutually Exclusive Events

• When the sets corresponding to two events do not
  intersect, the events are said to be mutually
  exclusive. In this case,
• P(A I B) 0
• Therefore, for mutually exclusive events the rule
  for unions reduces to
• P(A U B) P(A) P(B)

45
Mutually Exclusive Events

• Two events are said to be mutually exclusive if
  the events have no sample points in common. That
  is, two events are mutually exclusive if, when
  one event occurs, the other cannot occur.

Sample Space S
Event B
Event A
46
Example of Unions with Mutually Exclusive Events

• If you draw one card, what is the probability of
  drawing either a heart or a spade?
• Are the two events, heart-spade, mutually
  exclusive? Can they occur simultaneously?
• P(A U B) P(A) P(B)
• P(A UB) P(A) P(B) 13/52 13/52 1/2

47
General Addition Rule Union of Two Events

• The union of events A and B is the event
  containing all sample points that are in A or B
  or both.
• The union is denoted by A ??B?
• The union of A and B is illustrated below.

Sample Space S
Event A
Event B
48
Rule of Unions

• P(A U B) P(A) P(B) - P(A I B)
• where
• U union of two events
• I intersection of two events

This is sometimes referred to as the Addition
Rule for events that are not mutually exclusive.
49
Union Interpretation

• P(A U B) P(A) P(B) - P(A I B)
• A union of two events measures the occurrence of
  A or B. However, if we add the separate
  probabilities as the rule implies we will double
  count the elements that are both in A and B.
  Thus, we subtract the intersection of A and B.

50
Not Mutually exclusive

• A card is drawn at random from an ordinary deck
  of cards. Find the probability that a card is a 5
  or a Club.
• Now 5 and C Are not mutually exclusive. So
• P(5?C)P(5)P( C )-P(5?C)1/131/4-1/524/13

51
Union Example

• Drink Dont Drink
• Under Age 30 10 40
• Of age 50 10 60
• 80 20
• What is the probability of under age or dont
  drink?

52
Union Example
Drink Dont Drink Under Age 30 10
40 Of age 50 10 60 80 20
100 What is the probability under age(UA) or
dont drink(DD)? P(UA U DD) P(UA) P(DD) -
P(UA I DD) 40/100 20/100 - 10/100
50/100 0.5
53
Probability to Odds

• If P(A) is the probability of an event A
  occurring, then
• the odds in favor of A
• P(A)?1
• The odds in against A
• P(A)?0

54
Example

• The probability of event A occurring is 0.8
• A. Determine the odds if favor of A occurring
• B. Determine the odds against A occurring

55
Odds to Probability

• If the odds for event A are a to b then the
  probability of event A occurring is

56
Example

• The odds in favor of event A occurring are 5 to
  3.
• Determine the probability.

57
Computing Probabilities Using the Multiplication
Rule

• Independent Events
• Conditional Probability
• General Multiplication Rule

58
Independent Vs Dependent

• If an event has no effect on the occurrence of
  another event these events are called independent
  events.
• If two events are not independent they are
  dependent events.
• Two Events are independent if any of the
  following holds
• P(AB)P(A)
• P(BA)P(B)
• P(A?B)P(A)P(B)
• Otherwise the events are said to be dependent.

59
Independent Vs Dependent

• Consider the following events in the toss of a
  single die
• A Observe an even number.
• B Observe an odd number
• C Observe a 5 or a 6
• Are A and B independent events ?
• Are A and C independent events ?

60
Continued

• a. Let s see if A and B satisfy the conditions
  in slide 43.
• P(A)1/2, P(B)1/2, P(C)2/6.
• P(A?B)0 because or we get even or odd !! And
  P(A)P(B)1/4 therefore A and B are dependent
  events.
• b. P(A ?C)1/6 and P(A)P(C)1/6 therefore A and
  C are independent.

61
Intersection of Two Events

• The intersection of events A and B is the set of
  all sample points that are in both A and B.
• The intersection is denoted by A ????
• The intersection of A and B is the area of
  overlap in the illustration below.

Sample Space S
Intersection
Event A
Event B
62
Intersections for Independent EventsMultiplicatio
n Rule
If A and B are independent events then P(A and
B)
P(A I B) P(A)P(B)

• Conditions for independence of two events A an B
• P(AB) P(A)
• P(BA) P(B)

63
Intersection Rule for Dependent Events General
Multiplication Rule

• P(A I B) P(AB)P(B)
• or
• P(A I B) P(BA)P(A)

This is sometimes called the General
Multiplication Rule for dependent events
64
Conditional Probability

• The probability of an event given that another
  event has occurred is called a conditional
  probability.
• The conditional probability of A given B is
  denoted by P(AB).
• A conditional probability is computed as follows

P(B) gt 0
P(A) gt 0
65
Conditional Probability Example

• What is the probability that a single toss of a
  die will result in a number less than 4 if it is
  given that the toss resulted in an odd number. If
  B is the event less than 4and if A odd
  number)1/2 then P(A I B)2/6 (that is 1 and 3)
• P(A I B) P(A I B) 1/3
• P(BA) --------------- P(BA)
  -------------- ------ 2/3
• P(A) P(A) 1/2

1 2 3 4 5
6
Sample Space for Roll of Die
66
Bayes Theorem

• Simple application of conditional probability
• recognized in 1761 by Reverend Thomas Bayes
• used as a foundation for a new philosophy of
  science
• revise probabilities in response to new
  information

67
Bayes Theorem

• Often we begin probability analysis with initial
  or prior probabilities.
• Then, from a sample, special report, or a product
  test we obtain some additional information.
• Given this information, we calculate revised or
  posterior probabilities.
• Bayes theorem provides the means for revising
  the prior probabilities.

68
Bayes Theorem

• To find the posterior probability that event Ai
  will occur given that event B has occurred we
  apply Bayes theorem.
• Bayes theorem is applicable when the events for
  which we want to compute posterior probabilities
  are mutually exclusive and their union is the
  entire sample space.

69
Continued

• 2nd

70
Example

• An automobile dealer has kept records on the
  customers who visited his showroom. 40 of the
  people who visited his dealership were female.
  Furthermore, his records show that 35 of the
  females who visited his dealership purchased an
  automobile, while 20 of the males who visited
  his dealership bought an automobile.
• Let A1 the event that the customer is female
• A2 the event the customer is male

71
Example

• What is the probability that a customer entering
  the showroom will buy a car ?
• A car sales person has just told us that he sold
  a car to a customer. What is the probability that
  the customer was female ?
• Prepare a table in order to calculate
  P(A1B),P(A2B) and P(B)
• Draw a complete probability tree for the above
  problem

72
(A)

• What is the probability that a customer entering
  the showroom will buy a car ?
• In this case we want to determine the probability
  that a customer, regardless of the gender of the
  customer, will buy a car. Let B the event that
  the customer will buy a car. We know that
• P(BA1) 0.35 that is the probability that a
  customer will buy a car under the condition that
  the customer is female is 0.35
• P(BA2)0.20 that is the probability that a
  customer will buy a car under the condition that
  the customer is male is 0.20

73
(D)

• (d)Draw a complete probability tree for the above
  problem

Buy A Car
Probability
Event
BA1
Buy
A1?B
Yes
.14
.35
Female
BA1
.26
A1?B
No
.65
Not Buy
.4
.6
.20
Buy
A2?B
Yes
.12
Male
.80
No
.48
A2?B
Not Buy
74
(A) continued

• The probability that a customer will buy a car,
  that is P(B) can be computed as
• P(B)P(A1)P(BA1)P(A2)P(BA2)(.4)(.35)(.6)(.2)
  .26
• This indicates that the probability of a customer
  buying a car regardless of the customers gender
  is 0.26.

75
(B)

• (b) A car sales person has just told us that he
  sold a car to a customer. What is the probability
  that the customer was female ?
• Based on the info that a purchase was made, we
  can revise the probability of a customer being a
  female.

76
(B) Continued

• Substituting the values in the previous equation
  we have
• Hence the probability that the customer was a
  female is revised from 0.4 to 0.538

77
( C )

• ( c )Prepare a table in order to calculate
  P(A1B),P(A2B) and P(B)
• (1) (2) (3)
  (4) 5)
• Prior Conditional
  Joint Posterior
• Events Probabilities Probabilities
  Probabilities Probabilities
• Ai P(Ai) P(BAi)
  P(Ai I B) P(Ai B)
• A1 .4 .35 .14 .14/.26.538
• A2 .6 .20 .12 .12/.26.462
• 1.0 P(B) .26 1.00