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					 June 16           Day 6
                The Addition
                                  Read pp. 274 - 281                                 due
    5.2           Rule and
                                  Do pp. 281 - 285, # 1 – 47 (odd)                 June 21
                Complements
                 Vocabulary: { disjoint, mutually exclusive, contingency table, row variable,
                             column variable, cell, complement of an event }
                Independence
                   and the        Read pp. 286 - 289                                 due
    5.3
                Multiplication    Do pp. 290 - 292, # 1 - 31                       June 21
                    Rule
                  Vocabulary: { disjoint events, independent events, At-least probabilities }

Addition Rule for Disjoint Events

          Two events are disjoint if they have no outcomes in common. Another name for
          disjoint events is mutually exclusive events.




What does “S” stand for?




                                                                                                1
Example: Using the Venn Diagram on Page 1:

   (A)         P(Red) =


   (B)         P(0) =




Addition Rule for Disjoint Events


If E and F are disjoint (or mutually exclusive) events, then

                             P  E or F   P  E   P  F 



note: The Addition Rule also works with more than two mutually exclusive events.



Example:                Benford‟s Law

When we write numbers, we do not write them with a “0” in the first digit, i.e., 25 is not
025. Thus, numbers begin with 1 through 9. We might expect that the first digit of the
numbers we use would begin with equal frequency, all numbers appearing 1/9 of the
time. Physicist Frank Benford was credited with stating that they do not. In fact, he
asserted that they occur as shown in the table below:




                                                                                             2
   (A)          Verify that Benford‟s Law is a probability model



   (B)          Use Benford‟s Law to determine the probability that a randomly selected
                first digit is 3 or 6



   (C)          Use Benford‟s Law to determine the probability that a randomly selected
                number is greater than 5.




Example # 2a:          Using a Standard Deck of Cards (52 and no Jokers)

   (A)          Compute the probability of the event E = “drawing a 9”



   (B)          Compute the probability of the event E = “drawing a 9” or “drawing a 3”



   (C)          Compute the probability of the event E = “drawing a 9” or “drawing a 3”
                or drawing a Queen”




Example #2b: If A and B are mutually exclusive events with P(A) = .33 and P(B) = .25,
             what is the P(A or B)?




                                                                                          3
The General Addition Rule

For any two events E and F:

                    P  E or F   P  E   P  F   P  E and F 



Example #3a: A two-question survey was taken of Tri-C.
             “Do you like chocolate ice cream?” and “Do you like vanilla ice cream?”




                      E = Chocolate                 F = Vanilla


                      P  E   42%
                      P  F   20%
                      P  E and F   14%

                     Find P  E or F 




                                                                                   4
Example #3b:

The registrar‟s office searched student records to find what percent of the freshman were
enrolled in a math class. A second search discovered what percent of the freshman were
taking an English class.




                 M = math class                             E = English class

                       P  M   .66
                       P  M or E   .92
                       P  M and E   .48

                      Find P  E 

Example #3c:

The guidance department totaled the results from Freshman surveys. The two largest
areas of career interests were medical and business.




                       M = medical                      B = business

                                       P  M   .23
                                       P  B   .25
                                       P  M or B   .34

                                     Find    P  M and B 




                                                                                            5
Contingency Tables (or two-way tables)

Example #4:

A furniture store did a survey of 180 customers regarding customer satisfaction as well as
how the customer learned of the store.

                        TV ad            Walk-In            Referred           Total
Not Satisfied            5                  10                4                19
Neutral                  12                 9                 19               40
Satisfied                22                 18                28               68
Very Satisfied           15                 14                24               53
Total                    54                 51                75               180
Assume the sample represents the entire population of customers. Find the probability
that a customer is:


   a)       Learned of the store through a TV ad.


   b)       Very satisfied


   c)       Arrived at the store as a Walk-In or Referred


   d)       Was Satisfied or Referred


   e)       Satisfied and Referred




                                                                                         6
    The sum of the probabilities of all simple events in a sample space must equal 1.




    The complement of event A is the vent that A does not occur. Ac designates the
                       complement of event A. Furthermore,

       1.      P  E   P  Ec   1
       2.      P  event E does not occur   P  E c   1  P  E 

Example #6a: Tri-C has renovated one of the classrooms and installed „captains‟ chairs
             that are both cushioned and swivel. The interior designer also thought it
             would be appealing to install chairs with different colors. In one
             classroom, the following colored chairs were installed:

               Green          Blue          Red          Yellow         Black
                 8             4             6             7              5

              The professor announces that he will select a color and then all students
              who are seated in those colored chairs will have to make a presentation
              from the homework. You are seated on a green chair and are worried
              because last night you went to the Indians game and didn‟t do your
              homework. What is the probability that you will have to make a
              presentation?



              What is the probability that you won‟t have to make a presentation?




Example #6b: Playing roulette.

        The primary pockets on a Roulette wheel are numbered from 1 to 36 with each
alternating between red and black. There also is a green 0 and in the American Roulette
another green 00. Using the Law of Large numbers, if you played the game 2000 times
selecting only RED on each play, how many times would you expect to win?




                                                                                          7
               Independence
                  and the        Read pp. 286 - 289                                 due
    5.3
               Multiplication    Do pp. 290 - 292, # 1 - 31                       June 21
                   Rule
                 Vocabulary: { disjoint events, independent events, At-least probabilities }

Two events are independent if the occurrence or nonoccurrence of one does not change
the probability that the other will occur.

Two event are dependent if the occurrence or nonoccurrence of one does affect the
probability that the other will occur.

Example # 1             Independent or Not?

   A) A fair coin is tossed four times and it comes up HEADS all four times. Your
      friend says that the next toss will most likely come up TAILS. Right or Wrong?


   B) A survey of people residing in Cuyahoga County in 2009 has several categories.
      Two of the categories are: “Attends Private College/University” and “Family
      Income greater than $100,000”. Are these independent?


   C) Twin sisters Eva and Deeva are 21. One auditions and is selected for the
      Cavaliers Dance Team and the other auditions and is selected for The Ohio State
      Buckeyes Cheer Team. Are these independent?


Disjoint Events are not the same as Independent Events

These may seem at first glance to be the same. However, two events are disjoint if they
have no outcomes in common. Independent events means that one does not affect the
other.

E = “Born in a month ending in Y”               F = “Born in a month not ending in Y”

E and F are disjoint.

J = “Mrs. Brown has a baby in July”
K = “Mrs. Brown‟s Grandmother goes on a cruise.”




                                                                                               8
Multiplication rule for independent events:

                             P  E and F   P  E   P  F 


Example #2a: A fair coin is tossed at the same time that a die is rolled. What is the
             probability that the coin will land heads up and the die will show a 4?




Example #2b: A fair coin is tossed at the same time that a die is rolled. What is the
             probability that a coin will land tails up and the die will show a number
             greater than 4?




The Multiplication rule can be extended to more than two events:

Multiplication rule for n independent events:

               P  E and F and G and ...  P  E   P  F   P G   ...




Example # 3: The Nevada Gambling Commission requires that odds for slot machines
must be posted on the actual machine. Usually they are posted on the back which is
pressed against a wall of the back of another slot machine. (It is strongly recommended
that you do not attempt to pull the slot machine out to verify this!) The odds vary. We
are about to play a slot machine in Casino del Luthy. The odds are 47%.

   a) What are the odds that you will win on the first try?



   b) What are the odds that you will win on both the second and third try?



   c) What are the odds that you would win on three consecutive tries?


                                                                                          9
Example 4: Black Jack or “21” is one of the very few games of chance where a smart
gambler can tip the odds in his favor. In order to do this, the gambler must be aware of
which cards have already been played and what are the probabilities for the cards still to
be dealt. In a deck of 52 cards, there are four Aces. What is the probability that both of
the first two cards dealt will be twos?




Example 5:     The tour director at the Smelly Jelly Bean Company was a former
               statistics teacher. He like to fill a ceramic jar (non-transparent) with
               candies and then let guests at the factory reach in and take out a candy.
               He then asked them which color had the largest number of beans. Below
               is a table of how he recently filled the jar.


Color           Blue         Green         Red         Yellow        Brown         Black
Number           31           24           40            17            50           33


       A guest will get to select one bean. The statistics teacher will replace whatever
       color is chosen with the same color before the next guest has an opportunity.

       a)      What is P(red)?

       b)      What is P(green) followed by P(yellow)?

       c)      What is the P(brown) followed by P(black) and another P(brown)


The tour director has run out of replacements and no longer can put additional candies in
the jar.

       d)      What is P(blue) followed by P(green)?

       e)      What is P(black) followed by P(black)?




                                                                                           10
Computing At-Least Probabilities:

The concept at play with “At-Least” is the complement rule.



    The complement of event A is the vent that A does not occur. Ac designates the
                       complement of event A. Furthermore,

       1.      P  E   P  Ec   1
       2.      P  event E does not occur   P  E c   1  P  E 




Example # 4: Mrs. Tell, expert bow-and-arrow marksman, hits her target 95% of the
time from a distance of 50 feet. Suppose one day she is baby-sitting the six Brady-Bunch
children and decides to play “Shoot the Apple Off Your Head”. The eager children
willingly line up 50 feet away with apples on their head. What is the probability that at
least 1 time she will miss the apple?


Method 1:

P(1 or 2 or 3 or 4 or 5 or 6) = P (1 miss) + P (2 misses) + P (3 misses) + …

This is time consuming and difficult. In fact, we really don‟t have the mathematical
background at this time to make the calculations. We will learn about it in Lesson 6.2.

Method 2:

P (at least one miss) =




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