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4-3 Counting Techniques

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					4.3 Counting
Techniques
     Prob & Stats
Tree Diagrams

   When calculating probabilities,
    you need to know the total
                 outcomes
    number of _____________ in
    the ______________.
         sample space
Tree Diagrams Example
   Use a TREE DIAGRAM to list the
    sample space of 2 coin flips. Sample
                                                      Space

                    H
                                            H
                                Now you could get…
                                   If you got H
                                            T


        On the first flip you
          could get…..
YOU


                                          H
                                    you could
                                NowIf you got Tget…
                                           T


                    T
Tree Diagram Example
   Mr. Arnold’s Closet

            3 Shirts            2 Pants




             2 Pairs of Shoes
Dress Mr. Arnold
   List all of Mr. Arnold’s outfits   1

                                       2
Dress Mr. Arnold
   List all of Mr. Arnold’s outfits   1

                                       2
                                           3

                                           4
Dress Mr. Arnold
   List all of Mr. Arnold’s outfits       1

                                           2
                                               3

                                               4

                                       5

                                       6
Dress Mr. Arnold
   List all of Mr. Arnold’s outfits       1

                                           2
                                               3

                                               4

                                       5

                                       6
                                                   7


                                                   8
Dress Mr. Arnold
   List all of Mr. Arnold’s outfits       1

                                           2
                                                    3

                                                   4

                                       5

                                       6
                                                        7


                                                        8

                                               9
                                               10
Dress Mr. Arnold
   List all of Mr. Arnold’s outfits       1

                                           2
                                                    3

                                                   4

                                       5

                                       6
                                                        7


                                                        8

                                               9
                                               10

                                                       11

                                                       12
Dress Mr. Arnold                                            If Mr. Arnold picks an
                                                               outfit with his eyes
   List all of Mr. Arnold’s outfits       1                           closed…….

                                           2                P(brown shoe) =
                                                    3
                                                                 6/12
                                                                 1/2
                                                   4
                                                             P(polo) =
                                       5

                                       6
                                                                   1/3
                                                                   4/12
                                                        7
                                                              P(lookin’ cool) =

                                                        8

                                               9
                                               10

                                                       11
                                                                      1
                                                       12
Multiplication Rule of Counting

   The size of the sample space is
        denominator
    the ___________ of our
    probability
   So we don’t always need to
    know what each outcome is, just
    the number of outcomes.
Multiplication Rule of
Compound Events
If…
 X = total number of outcomes
       for event A
 Y = total number of outcomes
       for event B
 Then number of outcomes for A
  followed by B = x times y
                   _________
Multiplication Rule:
Dress Mr. Arnold

   Mr. Reed had 3 EVENTS


          2
         shoes
                     2
                   pants
                                3
                              shirts




          How many outcomes are there
              for = 12 OUTFITS
          2(2)(3)EACH EVENT?
Permutations

   Sometimes we are concerned
    with how many ways a group of
                    arranged
    objects can be __________.
      •How many ways to arrange books on a
      shelf

      •How many ways a group of people can
      stand
       in line

      •How many ways to scramble a word’s
      letters
                                        Wonder Woman’s invisible plane has 3
Example:                             
                                         chairs.
                                         There are 3 people who need a lift.
3 People, 3 Chairs                      How many seating options are there?
                            Think of each chair as
 Wonder Woman driving     6 Seating Options!
                                 an EVENT
                                                          Batman driving



                          3          2        1
                           Now that ways is filled?
                          How manythe 1st could the
                            Now the first 2 are filled.
                           How = options for
                              many 6 OPTIONS
                        3(2)(1)many ways to fill ndrd?
                               st
                          How 1 chair be filled? 2 3 ?

                         Superman driving
Example:                  
                          
                              The batmobile has 5 chairs.
                              There are 5 people who need a lift.
5 People, 5 Chairs           How many seating options are there?




      5     4       3         2       1         =120
                                                Seating Options



                Multiply!!
           This is a PERMUTATION of 5 objects
Commercial Break:
FACTORIAL

   denoted with !  5!
   Multiply all integers ≤ the number 
         5! = 5(4)(3)(2)(1) = 120
   0! = 1
   1! = 1
   Calculate 6!
      6! = 6(5)(4)(3)(2)(1) = 720
   What is 6! / 5!?
Commercial Break:
FACTORIAL

   denoted with !  5!
   Multiply all integers ≤ the number 

   0! = 1
   1! = 1
   Calculate 6!

   What is 6! / 5!?
      6(5)(4)(3)(2)(1)
        5(4)(3)(2)(1)
                                  =6
Example:                  
                          
                              The batmobile has 5 chairs.
                              There are 5 people who need a lift.
5 People, 5 Chairs           How many seating options are there?




                                                   5!
      5     4       3         2       1         =120
                                                Seating Options



                Multiply!!
           This is a PERMUTATION of 5 objects
Permutations:
Not everyone gets a seat!
   It’s time for annual Justice League softball game.
   How many ways could your assign people to play 1st,
    2nd, and 3rd base?
                                                     What if I
                                                   choose these
                                                        3?




               You have to choose 3 AND             Think of the
                                                    possibilities!
                     arrange them
Permutations:
Not everyone gets a seat!
   It’s time for annual Justice League softball game.
   How many ways could your assign people to play 1st,
    2nd, and 3rd base?
                                                     What if I
                                                   choose these
                                                        3?




               You have to choose 3 AND             Think of the
                                                    possibilities!
                     arrange them
Permutations:
Not everyone gets a seat!
   It’s time for annual Justice League softball game.
   How many ways could your assign people to play 1st,
    2nd, and 3rd base?
                                                     What if I
                                                   choose these
                                                        3?




               You have to choose 3 AND             Think of the
                                                    possibilities!
                     arrange them
Permutations:
Not everyone gets a seat!
   It’s time for annual Justice League softball game.
   How many ways could your assign people to play 1st,
    2nd, and 3rd base?                                 BUT…
                                                    What if I
                                                    choose
                                                   THESE 3?




               You have to choose 3 AND            Think of the
                                                   possibilities!
                     arrange them
Permutations:
Not everyone gets a seat!
   It’s time for annual Justice League softball game.
   How many ways could your assign people to play 1st,
    2nd, and 3rd base?                                 BUT…
                                                    What if I
                                                    choose
                                                   THESE 3?




               You have to choose 3 AND            Think of the
                                                   possibilities!
                     arrange them
Permutations:
Not everyone gets a seat!
   It’s time for annual Justice League softball game.
   How many ways could your assign people to play 1st,
    2nd, and 3rd base?                                 BUT…
                                                    What if I
                                                    choose
                                                   THESE 3?




               You have to choose 3 AND            Think of the
                                                   possibilities!
                     arrange them
Permutations:
Not everyone gets a seat!
   It’s time for annual Justice League softball game.
   How many ways could your assign people to play 1st,
    2nd, and 3rd base?                                 BUT…
                                                    What if I
                                                    choose
                                                   THESE 3?



                This is going to
                      take
             FOREVER
               You have to choose 3 AND            Think of the
                                                   possibilities!
                     arrange them
You have 3 EVENTS?
   How many outcomes for each event




                                  How many
                               outcomes for this
                                    event!


                                      5


           You have to choose 3 AND
                 arrange them
You have 3 EVENTS?
                How many
             outcomes for this
                  event!



                    4
                                 Now someone is
                                   on FIRST




                                        5

       You have to choose 3 AND
             arrange them
You have 3 EVENTS?
    5(4)(3) = 120 POSSIBLITIES
                    And on SECOND



                             4
                                    Now someone is
                                      on FIRST



                3                          5
   How many
outcomes for this
     event!


               You have to choose 3 AND
                     arrange them
Permutation Formula

    You have n objects
    You select r objects
    This is the number of ways you
     could select and arrange in
     order:




    Another common notation for a permutation is nPr
Softball Permutation Revisited
                         5!
                     5(4)(3)(2)(1)
     5 people to choose from
 n=
 r = 3 spots to fill (5 – 2!
                          3)!
                        2(1)

    5(4)(3) = 120 POSSIBLITIES




        You have to choose 3 AND
              arrange them
    Combinations

   Sometimes, we are only
    concerned with selecting a group
    and not the order in which they
    are selected.
   A combination gives the number
    of ways to select a sample of r
    objects from a group of size n.
Combination: Duty Calls
   There is an evil monster threatening
    the city.
   The mayor calls the Justice League.
   He requests that 3 members be sent
    to combat the menace.
   The Justice League draws 3 names
    out of a hat to decide.
   Does it matter who is selected first?
                 NOPE
   Does it matter who is selected last?
               NOPE
Combination: Duty Calls
  Let’s look at the drawing possibilities




                           STOP!
                     This is a waste of
                            time
          These are all them
          We’ll countthewho gotas
       The monster doesn’t care
                                SAME:
                                   drawn
            ONE OUTCOMEfirst.
          All these outcomes = same people
                                  We’ll count them as
                                 These are all the SAME:
                   pounding his face
                            The monster doesn’t care who got drawn
                                   ONE OUTCOME
                                            first.
                                  All these outcomes = same people
                                           pounding his face
Combination: Duty Calls

            Okay, let’s consider other outcomes




          10 Possible Outcomes!
Combination Formula

   You have n objects
   You want a group of r objects
   You DON’T CARE what order
    they are selected in




       Combinations are also denoted nCr
               Read “n choose r”
Duty Calls: Revisited
   n = 5 people toDOESN’T MATTER
         ORDER
                     choose from
   r = 3 spots to fill

               5(4)(3)(2)(1)
                   5!                  20
                                       2
                 3!(2)!
               3!(5 - 3)!
               3(2)(1)(2)(1)

           10 Possible Outcomes!

         Now we can go save the city
    Permutation vs. Combination

   Order matters  Permutation
   Order doesn’t matter Combination

				
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