Probability Rules.ppt

Document Sample
Probability Rules.ppt Powered By Docstoc
					Probability Rules
    Chapter 15




                    1
Review of Approaches to Probability
1.) What is the probability that the Dow Jones Industrial
  Average will exceed 12,000? Which approach to
  probability would you use to answer this question?
2.) The National Center for Health Statistics reports that of
   883 deaths, 24 resulted from an automobile accident, 182
   from cancer, and 333 from heart disease. What is the
   probability that a particular death is due to an automobile
   accident? Which approach to probability did you use to
   answer this question?
3.)One card will be randomly selected from a standard 52-
   card deck. What is the probability the card will be an Ace?
   Which approach to probability did you use to answer this
   question?

                                                                 2
Review of Basic Probability Rules
Complement Rule
     P(A) = 1 – P(Ac)
Addition Rule for Mutually Exclusive Events
     P(A or B) = P(A) + P(B)
Multiplication Rule for Independent Events
     P(A and B) = P(A)*P(B)
“At Least One” Rule
     P(At least one) = 1 – P(none)
                                              3
Venn Diagram:
Mutually Exclusive Events
 One card is drawn from a standard deck of
 cards. What is the probability that it is an
 ace or a nine?




                       A           B


                 Ace                    nine


 Events A and B are mutually exclusive. A card can be either
 an Ace or a nine, but can not be both                         4
Venn Diagram:
Events that are Not Mutually Exclusive
 One card is drawn from a standard deck of
 cards. What is the probability that it is red or
 an ace?



                      Red          Ace


                      Both red and an ace


  These events are not mutually exclusive as it is possible for
  a card to be both red and an ace (ace of hearts, ace of
  diamonds)                                                       5
Probabilities of Events that Are Not
Mutually Exclusive
 Find P(Red or Ace):



                    0.5        0.077
                   Red          Ace

                            0.038


     = P(Red) + P(Ace) – P(Both Red and Ace)
     = 0.5 + 0.077 -0.038
     =0.539
                                               6
General Addition Rule
Used when events are not mutually exclusive.


       P(A or B) = P(A) + P(B) – P(A and B)


Example: The Illinois Tourist Commission selected a
sample of 200 tourists who visited Chicago during the
past year. The survey revealed that 120 tourists went
to the Sears Tower, 100 went to Wrigley Field and 60
visited both sites. What is the probability of selecting a
person at random who visited both the Sears Tower and
Wrigley Field?

                                                             7
General Addition Rule (continued)
P(Sears Tower) = 120/200 = 0.6
P(Wrigley Field) = 100/200 = 0.5
P(Both) = 60/200 = 0.3


                             0.3

                     0.6           0.5
                         S         W



   P(S or W) = P(S) + P(W) – P(S and W)
             = 0.6   + 0.5   - 0.3
             = 0.8                        8
General Addition Rule (continued)
  What is the probability that a randomly selected person
  visited either the Sears Tower or Wrigley Field but NOT
  both?
  P(S or W but NOT both) = P(S or W) – P(S and W)
                = 0.8 – 0.3 = 0.5


  Second approach: P(S and Wc) = 0.6 – 0.3 = 0.3
                     P(W and SC) = 0.5 – 0.3 = 0.2
  P(S or W but NOT both) = P(S and Wc) + P(W and SC)
                        =   0.3   +   0.2   = 0.5

                                                            9
General Addition Rule (continued)
 What is the probability that a randomly
 selected tourist went to neither location?


 P(neither location) = 1 – P(either location)
                    = 1 – P(S or W)
                    = 1 – 0.8 = 0.2




                                                10
Conditional Probabilities
  Here is a contingency table that gives the counts of ECO 138
  students by their gender and political views. (Data are from Fall
  2005 Class Survey)

                              Political views
              Liberal        Moderate Conservative Total
Gender Male        17                29          14   60
       Female      30                24          23   77
       Total       47                53          37 137

   P(Female) = 77/137 = 0.562


   P(Female and Liberal) = 30/137 = 0.219
   What is the probability that a selected student has moderate       11
   political views given that we have selected a female?
Conditional Probabilities (continued)
 What is the probability that a selected student has moderate
 political views given that we have selected a female?

                            Political views
                 Liberal   Moderate Conservative Total
   Gender Male        17           29          14   60
          Female      30           24          23   77
          Total       47           53          37 137



  P(Moderate | Female) = 24/77 = 0.311


  Conditional probability, P (B|A) – the probability of event B
  given event A.
                                                                  12
Conditional Probabilities (continued)
 Formal Definition:
      P(B|A) = P(A and B)
                      P(A)
 Example: P(Moderate and Female)
                 P(Female)
            =(24/137) / (77/137)
            = 0.175 / 0.562
            = 0.311
                                    13
Multiplication Rule
 Multiplication Rule for Independent events:
        P(A and B) = P(A) * P(B)


 Independent – the occurrence of one event has no effect on
 the probability of the occurrence of another event.


 Example: A survey by the American Automobile Association
 (AAA) revealed that 60 percent of its members made airline
 reservations last year. Two members are selected at random.
 What is the probability both made airline reservations last
 year?
        P(R1 and R2) = P(R1)*P(R2) = (0.6)*(0.6) = .36

                                                               14
General Multiplication Rule

 Use when events are Dependent.


      P(A and B) = P(A) * P(B|A)


 For two events A and B, the joint probability
 that both events will happen is found by
 multiplying the probability event A will happen
 by the conditional probability of event B
 occurring.
                                                   15
General Multiplication Rule (continued)
Example: A county welfare agency employs
10 welfare workers who interview prospective
food stamp recipients. Periodically the
supervisor selects, at random, the forms
completed by two workers to audit for illegal
deductions. Unknown to the supervisor, three
of the workers have regularly been giving
illegal deductions to applicants. What is the
probability that both of the two workers
chosen have been giving illegal deductions?
                                          16
General Multiplication Rule (continued)
Solution: Define the following two events:
  A = First worker selected gives illegal deductions
  B = Second worker selected gives illegal deductions
We want to find the probability that both A and B occur.

To find the P(A) consider the following Venn Diagram.
I = worker with illegal deductions N = worker not giving illegal D


 N1 N2 N3                          Each observation in the sample
                    I1        I2
                                   space is equally likely.
 N4 N5 N6                I3
                A                  P (A) = P(I1) + P(I2) + P(I3)
     N7
                                        = 1/10 + 1/10 + 1/10
                                        = 3/10 or 0.30              17
General Multiplication Rule (continued)
To find the conditional probability, P(B|A), we need to make changes
to the sample space. Remember our assumption is that the first
worker selected is giving illegal deductions.

                                  P (B|A) = P(I1) + P(I2)
    N1 N2 N3                             = 1/9 + 1/9 = 2/9
    N4 N5 N6        I1
       N7                         Substituting P(A) and P(B|A) into
                         I2
                  B|A             the formula for the general
                                  multiplication rule, we find
                                  P(A and B) = P(A)P(B|A)
                                         = (3/10) * (2/9)
                                         = 6/90 = 1/15 or 0.067

                                                                      18
Tree Diagram
A tree diagram is a display of conditional events or probabilities
that is helpful in thinking through conditioning.


                      N (6/9)       N and N = (7/10)(6/9) = 42/90


        N
            (7/10)
                        I = (3/9)
                                     N and I = (7/10)(3/9) = 21/90

            (3/10)   N =(7/9)
                                     I and N = (3/10)(7/9) = 21/90
        I


                     I = (2/9)       I and I = (3/10)(2/9) = 6/90
                                                                     19
Independent Events?
Again, events are independent when the outcome of one
event does not influence the probability of the other.


Events A and B are independent whenever
              P(B|A) = P(B)


In the case of independent events the general
multiplication rule reduces to the simple multiplication rule.
       P(A and B) = P(A) * P(B|A)
       = P(A) * P(B)

                                                                 20
Exploring Independence
 Is the probability of being liberal independent
 of gender for ECO 138 students?
                            Political views
                 Liberal   Moderate Conservative Total
   Gender Male        17           29          14   60
          Female      30           24          23   77
          Total       47           53          37 137

In other words, does P(Liberal | Female) = P(Liberal)?
        P(Liberal|Female) = 30/77 = 0.39
        P(Liberal) = 47/137 = 0.343
Because these probabilities are not equal, we can be pretty sure
that liberal political views are not independent of the student’s   21
gender
Let’s Try Some Examples
 1.) Two cards are drawn without
 replacement. What is the probability they
 are both aces?


 2.) What is the probability of getting 5
 hearts in a row?


 3.) I draw one card and look at it. I tell
 you that it is red. What is the probability it
 is a heart? And what is the probability it is
 red, given that it is a heart?                   22
Examples
 4.) Are “red card” and “spade” independent?
 Mutually exclusive?

 5.) Are “face card” and “king” independent?
 Mutually exclusive?




                                               23
Example - Travel
Suppose the probability that a U.S. resident has
traveled to Canada is 0.18, to Mexico is 0.09,
and to both countries is 0.04. What’s the
probability that an American chosen at random
has:
     A.) traveled to Canada but not Mexico?
     B.) traveled to either Canada or Mexico?
     C.) not traveled to either country?
     D.) Are travel to Mexico and Canada
        mutually exclusive events?
     E.) Are travel to Mexico and Canada independent   24
Example - Sick Cars
 Twenty percent of cars that are inspected have
 faulty pollution control systems. The cost of
 repairing a pollution control system exceeds
 $100 about 40% of the time. When a driver
 takes her car in for inspection, what’s the
 probability that she will end up paying more than
 $100 to repair the pollution control system?




                                                 25
Example - Health
The probabilities that an adult American man has high blood
pressure and/or high cholesterol are shown in the table:
                                     Blood Pressure
                Cholesterol          High      OK

                              High   0.11     0.21


                              OK     0.16     0.52

 A.) What is the probability that a man has both conditions?
 B.) What’s the probability that he has high blood pressure?
 C.) What’s the probability that a man with high blood
 pressure has high cholesterol?
 D.)Are high blood pressure and high cholesterol
 independent?                                                  26
Example - Absenteeism
A company’s records indicate that on any given
day about 1% of their day shift employees and
2% of the night shift employees will miss work.
Sixty percent of the employees work the day
shift.
     A.) Is absenteeism independent of shift
         worked? Explain.


     B.) What percent of employees are absent
         on any given day?                        27
Example - Blood Type
The American Red Cross says that about 45% of
the U.S. population has Type O blood, 40% Type
A, 11% Type B, and the rest Type AB.

Among four potential donors, what is the
probability that:

A.) All are Type O?
B.) No one is Type AB?
C.) They are not all Type A?
D.) At least one person is Type B?

                                                 28
Graduation
A private college report contains these statistics:
    70% of incoming freshmen attended public schools.
    75% of public school students who enroll as freshmen
    eventually graduate
    90% of other freshmen eventually graduate.


A.) Is there any evidence that a freshman’s chances to
graduate may depend upon what kind of high school the
student attended? Explain.


B.) What percent of freshman eventually graduate?

                                                           29
Assignment
   Read Chapter 16 (Random Variables) by
    Wednesday, March 8

   Try the following exercises
            #1,3,5,7,9,13,15,19,25,31, 35 and 43


   Quiz #3 – Wednesday, March 8
       Covers chapters 14 and 15
       Bring: pencil, calculator, UID card

                                                    30

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:0
posted:6/6/2012
language:
pages:30
liningnvp liningnvp http://
About