Chapter Five

1
Chapter Five
A Survey of Probability Concepts
GOALS When you have completed this chapter, you
will be able to
ONEDefine probability. TWO Describe the
classical, empirical, and subjective approaches
to probability. THREEUnderstand the terms
experiment, event, outcome, permutations, and
combinations. FOURDefine the terms conditional
probability and joint probability.
2
Chapter Five continued
A Survey of Probability Concepts
GOALS When you have completed this chapter, you
will be able to
FIVE Calculate probabilities applying the rules
of addition and the rules of multiplication. SIXU
se a tree diagram to organize and compute
probabilities. SEVEN Calculate a probability
using Bayes theorem.
3
Definitions

• A probability is a measure of the likelihood
  that an event in the future will happen.

• It it can only assume a value between 0 and 1.
• A value near zero means the event is not likely
  to happen. A value near one means it is likely.
• There are three definitions of probability
  classical, empirical, and subjective.

4
Definitions continued

• The classical definition applies when there are n
  equally likely outcomes.
• The empirical definition applies when the number
  of times the event happens is divided by the
  number of observations.
• Subjective probability is based on whatever
  information is available.

5
Definitions continued

• An experiment is the observation of some activity
  or the act of taking some measurement.
• An outcome is the particular result of an
  experiment.
• An event is the collection of one or more
  outcomes of an experiment.

6
Mutually Exclusive Events

• Events are mutually exclusive if the occurrence
  of any one event means that none of the others
  can occur at the same time.

• Events are independent if the occurrence of one
  event does not affect the occurrence of another.
  

7
Collectively Exhaustive Events

• Events are collectively exhaustive if at least
  one of the events must occur when an experiment
  is conducted.

8
Example 1

• A fair die is rolled once.
• The experiment is rolling the die.
• The possible outcomes are the numbers 1, 2, 3, 4,
  5, and 6.
• An event is the occurrence of an even number.
  That is, we collect the outcomes 2, 4, and 6.

9
EXAMPLE 2

• Throughout her teaching career Professor Jones
  has awarded 186 As out of 1,200 students. What
  is the probability that a student in her section
  this semester will receive an A?
• This is an example of the empirical definition of
  probability.
• To find the probability a selected student earned
  an A

10
Subjective Probability

• Examples of subjective probability are
• estimating the probability the Stormers will win
  the Super 12 this year.
• estimating the probability mortgage rates for
  home loans will drop to 12 percent.

11
Basic set theory

• A Set is a collection of objects of similar type
• Examples
• The set of all positive numbers
• All cricket players
• All companies listed on the JSE
• All bottles of red wine produced in 1980

12
Basic set theory

• The set of all outcomes is called the sample
  space or universe.

Venn diagram
A and B A n B
13
Basic set theory
A or B A U B
14
Basic Rules of Probability

• If two events A and B are mutually exclusive,
  the special rule of addition states that the
  probability of A or B occurring equals the sum of
  their respective probabilities
• P(A or B) P(A) P(B)

15
EXAMPLE 3

• SAA recently supplied the following information
  on their flights from Cape Town to Johannesburg

16
EXAMPLE 3 continued

• If A is the event that a flight arrives early,
  then P(A) 100/1000 .10.

• If B is the event that a flight arrives late,
  then P(B) 75/1000 .075.
• The probability that a flight is either early or
  late is
• P(A or B) P(A) P(B) .10 .075 .175.

17
The Complement Rule

• The complement rule is used to determine the
  probability of an event occurring by subtracting
  the probability of the event not occurring from
  1.

• If P(A) is the probability of event A and P(A)
  is the complement of A,
• P(A) P(A) 1 or P(A) 1 - P(A).

18
The Complement Rule continued

• A Venn diagram illustrating the complement rule
  would appear as

A
A
19
EXAMPLE 4

• Recall EXAMPLE 3. Use the complement rule to
  find the probability of an early (A) or a late
  (B) flight

• If C is the event that a flight arrives on time,
  then P(C) 800/1000 .8.
• If D is the event that a flight is canceled, then
  P(D) 25/1000 .025.

20
EXAMPLE 4 continued

• P(A or B) 1 - P(C or D)
• 1 - .8 .025 .175

D .025
C .8
(C or D) (A or B) .175
21
The General Rule of Addition

• If A and B are two events that are not mutually
  exclusive, then P(A or B) is given by the
  following formula
• P(A or B) P(A) P(B) - P(A and B)

22
The General Rule of Addition

• The Venn Diagram illustrates this rule

Avoid double counting
B
A and B
A
23
EXAMPLE 5

• In a sample of 500 students, 320 said they had a
  stereo, 175 said they had a TV, and 100 said they
  had both

TV 175
Both 100
Stereo 320
24
EXAMPLE 5 continued

• If a student is selected at random, what is the
  probability that the student has only a stereo,
  only a TV, and both a stereo and TV?

• P(S) 320/500 .64.
• P(T) 175/500 .35.
• P(S and T) 100/500 .20.

25
EXAMPLE 5 continued

• If a student is selected at random, what is the
  probability that the student has either a stereo
  or a TV in his or her room?

• P(S or T) P(S) P(T) - P(S and T)
• .64 .35 - .20 .79.

26
Joint Probability

• A joint probability measures the likelihood that
  two or more events will happen concurrently.
• An example would be the event that a student has
  both a stereo and TV in his or her dorm room.

27
Special Rule of Multiplication

• The special rule of multiplication requires that
  two events A and B are independent.

• Two events A and B are independent if the
  occurrence of one has no effect on the
  probability of the occurrence of the other.
• This rule is written P(A and B) P(A)P(B)

28
EXAMPLE 6

• Chris owns two stocks, IBM and General Electric
  (GE). The probability that IBM stock will
  increase in value next year is .5 and the
  probability that GE stock will increase in value
  next year is .7. Assume the two stocks are
  independent. What is the probability that both
  stocks will increase in value next year?
• P(IBM and GE) (.5)(.7) .35.

29
EXAMPLE 6 continued

• What is the probability that at least one of
  these stocks increase in value during the next
  year? (This means that either one can increase
  or both.)

• P(at least one) (.5)(.3) (.5)(.7) (.7)(.5)
• .85.

30
Conditional Probability

• A conditional probability is the probability of
  a particular event occurring, given that another
  event has occurred.
• The probability of the event A given that the
  event B has occurred is written P(AB).

31
General Multiplication Rule

• The general rule of multiplication is used to
  find the joint probability that two events will
  occur.

• It states that for two events A and B, the joint
  probability that both events will happen is found
  by multiplying the probability that event A will
  happen by the conditional probability of B given
  that A has occurred.

32
General Multiplication Rule

• The joint probability, P(A and B) is given by the
  following formula
• P(A and B) P(A)P(BA)
  or
  P(A and B) P(B)P(AB)

33
EXAMPLE 7

• The Dean of the School of Business at Wits
  University collected the following information
  about undergraduate students in her college

34
EXAMPLE 7 continued

• If a student is selected at random, what is the
  probability that the student is a female (F)
  accounting major (A)
• P(A and F) 110/1000.

• Given that the student is a female, what is the
  probability that she is an accounting major?
• P(AF) P(A and F)/P(F)
• 110/1000/400/1000 .275

35
Tree Diagrams

• A tree diagram is useful for portraying
  conditional and joint probabilities. It is
  particularly useful for analyzing business
  decisions involving several stages.

• EXAMPLE 8 In a bag containing 7 red chips and 5
  blue chips you select 2 chips one after the other
  without replacement. Construct a tree diagram
  showing this information.

36
EXAMPLE 8 continued
RR
6/11
7/12
R1
RB
5/11
BR
7/11
B1
5/12
BB
4/11
See example on p168
37
Bayes Theorem

• Bayes Theorem is a method for revising a
  probability given additional information.
• It is computed using the following formula

38
EXAMPLE 9

• Duff Cola Company recently received several
  complaints that their bottles are under-filled. A
  complaint was received today but the production
  manager is unable to identify which of the two
  Springfield plants (A or B) filled this bottle.
  What is the probability that the under-filled
  bottle came from plant A?

39
EXAMPLE 9 continued

• The following table summarizes the Duff
  production experience.

40
Example 9 continued
The likelihood the bottle was filled in Plant A
is reduced from .55 to .4783.
41
Some Principles of Counting

• The multiplication formula indicates that if
  there are m ways of doing one thing and n ways of
  doing another thing, there are m x n ways of
  doing both.

• Example 10 Dr. Delong has 10 shirts and 8 ties.
  How many shirt and tie outfits does he have?
• (10)(8) 80

42
Some Principles of Counting

• A permutation is any arrangement of r objects
  selected from n possible objects.
• Note The order of arrangement is important in
  permutations.

43
Some Principles of Counting

• A combination is the number of ways to choose r
  objects from a group of n objects without regard
  to order.

44
EXAMPLE 11

• There are 12 players on the Carolina Forest High
  School basketball team. Coach Thompson must pick
  five players among the twelve on the team to
  comprise the starting lineup. How many different
  groups are possible?
  

45
Example 11 continued

• Suppose that in addition to selecting the group,
  he must also rank each of the players in that
  starting lineup according to their ability.