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					           Chapter 6, Counting and Probability




CMSC 250                                         1
                                  Counting

      Counting elements in a list:
           – how many integers in the list from 1 to 10?
           – how many integers in the list from m to n? (assuming m <= n)




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           How many in a list divisible by something
      How many positive three-digit integers are there?
       – (this means only the ones that require 3 digits)
       – 999 – 100 + 1 = 900
      How many three-digit integers are divisible by 5?
       – think about the definition of divisible by
           x | y  ( k Z)[y = kx] then count the k’s that work
           100, 101, 102, 103, 104, 105, 106,…,994, 995, 996, 997, 998, 999



           20  5                    21  5    …       199  5
       – count the integers between 20 and 199
       – 199 – 20 + 1 = 180



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                             Probability
     The likelihood of a specific event.
     Sample space = set of all possible outcomes
     Event = subset of sample space
     Equal probability formula:
       – given a finite sample space S where all outcomes are equally
         likely
       – select an event E from the sample space S
       – the probability of event E from sample space S:
                    n( E )
           P( E ) 
                    n( S )



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                     Coins and cards and dice
          Two coins
           – sample space = {(H,H), (H,T), (T,H), (T,T)}
          Cards
           – values: 2,3,4,5,6,7,8,9,10,J,Q,K,A
           – suits: D(), H(), S(), C()
          Dice
           – sample space
             {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),
              (2,1),(2,2),(2,3),(2,4),(2,5),(2,6),
              …
              (6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}




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                      Multi-level probability
     If a coin is tossed once, the probability of head = ½
     If it’s tossed 5 times
                                       1 1 1 1 1      1
       – the probability of all heads:  * * * *  5
                                       2 2 2 2 2 2
                                           5
       – the probability of exactly 4 heads:
                                               25
     This is because the coin tosses are all independent
      events




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                     The breakfast problem
          Suppose your cereal can be Rice Krispies,
           cornflakes, Raisin Bran, or Cheerios.
          Suppose your drink can be coffee, orange juice, or
           milk.
          How many ways can you have breakfast?




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                    The multiplication rule
      If the 1st step of an operation can be performed n1 ways
      And the 2nd step can be performed n2 ways
      …
      And the kth step can be performed nk ways
      Then the operation can be performed n1n2  ∙ ∙ ∙  nk ways

      Cartesian product n(A)=3, n(B)=2, n(C)=4
       – n(A  B  C) = 24
       – n(A  B) = 6
       – n((A  B)  C) = 24




CMSC 250                                                         8
           Discrete Structures
               CMSC 250
               Lecture 34

              April 21, 2008




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           Using the multiplication rule for selecting
                             a PIN
           Number of 4 digit PINs of (0,1,2,.)
            – with repetition allowed = 4  4  4  4 = 256
            – with no repetition allowed = 4  3  2  1 = 24
           Extra rules :
            – . (the period) can’t be first or last
            – 0 can’t be first
                • with repetition allowed = 2  4  4  3
                • without repetition allowed = 2  2  2  1




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                      Probabilities with PINs
     Number of 4 digit PINs of {0,1,2,.}
       – with repetition allowed = 4  4  4  4 = 256
       – with no repetition allowed = 4  3  2  1 = 24
     What is the probability that your 4 digit PIN has no
      repeated digits?
     What is the probability that your 4 digit PIN does have
      repeated digits?

     Probability of the complement of an event
                  P(E’) = P(Ec) = 1  P(E)




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                   The difference rule formally
          If A is a finite set and B  A, then
                 n(A B) = n(A) – n(B)

          One application: probability of the complement of an
           event:
                     P(E’) = P(Ec) = 1  P(E)




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              PINs with less specified length-
                       addition rule

     Assume a PIN can be of length 2, 3, or 4, using {0,1,2,.}
     Partition the problem:
       – number of length-2 PINs w/rep allowed: 4  4 = 16
       – number of length-3 PINs w/rep allowed: 4  4  4 = 64
       – number of length-4 PINs w/rep allowed: 4  4  4  4 = 256


     Number of PINs if allowing length 2, 3, or 4 = 336




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                  The addition rule formally


      If A1  A2  A3 …  Ak = A
      and A1, A2 , A3,…,Ak are pairwise disjoint

   in other words, if these subsets form a partition of A, then

   n(A) = n(A1) + n(A2) + n(A3) + ∙∙∙ + n(Ak)




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               The inclusion/exclusion rule

     If there are two sets:
      n(A  B) = n(A) + n(B)
                – n(A  B)

     If there are three sets:
      n(A  B  C) =
        n(A) + n(B) + n(C)
        – n(A  B) – n(A  C) – n(B  C)
        + n(A  B  C)


CMSC 250                                      15
           Discrete Structures
               CMSC 250
               Lecture 35

              April 23, 2008




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                             Permutations

      Different ways of arranging all n of n objects
       – in either a line or a circle
       – without duplication/all items distinguishable
       – note: order is taken into account

      Number of linear permutations of N objects = N!
       N possible for 1st position  (N – 1) for 2nd  ∙ ∙ ∙  1 for last

      Number of circular permutations of N objects = (N  1)!
       Fix one person,
       then (N – 1) possible for next position * (N – 2) for 2nd  ∙ ∙ ∙  1 for
          last



CMSC 250                                                                           17
                            r-permutations
          If there are n things in a set, and you want to line up
           only r of them
                                 n!
           P(n, r )  n P r 
                              (n  r )!
          Example:
           Class = {Alice, Bob, Carol, Dan}
           – select a president and a vice president to represent the class
           – select a president, vice president, and webmaster




CMSC 250                                                                      18
                               Combinations
          Different ways of selecting objects
           – counting subsets
           – without duplication/all items distinguishable
           – note: order is not taken into account
                             0
            (n, r  Z where n  r )
                                                        
            C (n, r )    
                           n
                           r    
                                  P(n, r )
                                           
                                                 n!
                                             (n  r )! r!
                                   r!                   
      Examples: suppose {Alice, Bob, Carol, Dan} are on the
        ballot
           – select two superdelegates
           – select three superdelegates




CMSC 250                                                      19
      Permutations but of indistinguishable items
     Examples:
       – Arrangements of the word “baboons”
       – Assume you have a set of 15 beads:
           • 6 green
           • 4 orange
           • 3 red
           • 2 black
         select positions of the green ones, then the orange ones, then the
         red ones, then the black ones (or a different order of selecting their
         positions would work as well)

           
           15
           6    *  *  *    6!4!3!2!
                   9
                   4
                        5
                        3
                             2
                             2
                                     15!


CMSC 250                                                                          20
           Discrete Structures
               CMSC 250
               Lecture 36

              April 25, 2008




CMSC 250                         21
              Combinations with repetition
    {a,b,c,d,e}
    How many 3-combinations can be made without
     repetition?
    How many 3-combinations can be made with unlimited
     repetition allowed?

    These are multisets [a,b,c]
      – not sets {a,b,c}
      – and not tuples (a,b,c)


    How many combinations of 20 a's, b's, and c's can e
     made with unlimited repetition allowed?
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                      Notice similarities
    The number of nonnegative integer solutions of the
     equation
                                                           1i  n
     x1  x2      xn  r             xi  0, i  Z
    The number of selections, with repetition, of size r from a
     collection of size n.

    The number of ways r identical objects can be distributed
     among n distinct containers.




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           Choosing r elements out of n elements



                                order matters       order doesn’t matter

                                        n  r  1
                            n  n  n 
                               
                                               
               repetition                       r
                 allowed                         
                              r times
                                            r    
                            P ( n, r ) 
                                            n!      n      n!
                                                    
                                         (n  r )!  r  (n  r )! r!
           repetition not
                allowed                             

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                Probability with combinations
     Assume:
       – there are 32 people in a class
       – seven will be chosen to get extra homework
     What is the probability that you get extra homework?

     Number of ways to select the lucky seven
     Number of ways to select if you get homework
     P(you get homework)




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                         Tournament play
      Team A and Team B compete in a “best of 3” tournament
      They each have an equal likelihood of winning each game




       –   Do the leaves add up to 1?
       –   Do they always have to play 3 games?
       –   What's the probability the tournament finishes in 2 games?
       –   Do A and B have an equal chance of winning?

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             What if A wins 2 of every 3 games?
          Each line for A must have a 2/3
          Each line for B must have a 1/3




           – How likely is A to win the tournament?
           – How likely is B to win the tournament?
           – What is the probability the tournament finishes in two
             games?

CMSC 250                                                              27
      Where the multiplication rule doesn’t work

     People= {Alice, Bob, Carolyn, Dan}
     Need to be appointed as president, vice-president, and
      treasurer, and nobody can hold more than one office
       – how many ways can it be done with no restrictions?
       – how many ways can it be done if Alice doesn’t want to be
         president?
       – how many ways can it be done if Alice doesn’t want to be
         president, and only Bob and Dan are willing to be vice-
         president?




CMSC 250                                                            28
           Discrete Structures
               CMSC 250
               Lecture 37

              April 28, 2008




CMSC 250                         29
   Harder examples of selecting representatives

          Candidates= {Azar, Barack, Clinton, Dan, Erin, Fred}
      1.   select two, with no restrictions
      2.   select two, assuming that Azar and Dan must stay
           together
      3.   select three, with no restrictions
      4.   select three, assuming that Azar and Dan must stay
           together
      5.   select three, assuming that Barack and Clinton
           refuse to serve together



CMSC 250                                                          30
            Properties of combinations and their
                           proofs

                   1
                   n
                   0           
                                n 1
                                n

                   n
                  n
                  1             
                                n 1 
                                n
                                         n
           
           n
           2    
                  n(n  1)
                     2
                                
                                n
                                n2   
                                        n(n  1)
                                           2

                          
                       n
                       r
                              n
                              nr
CMSC 250                                           31
                       The binomial theorem
  ( x  y)  x  y
  ( x  y ) 2  ( x  y )( x  y )  x 2  xy  xy  y 2  x 2  2 xy  y 2
  ( x  y)3  ( x  y) 2 ( x  y)  ( x 2  2 xy  y 2 )(x  y ) 
        x 3  2 x 2 y  xy 2  x 2 y  2 xy 2  y 3  x 3  3x 2 y  3xy 2  y 3
  ( x  y ) 4  ( x  y ) 3 ( x  y )  ( x 3  3 x 2 y  3 xy 2  y 3 )( x  y ) 
        x 4  3 x 3 y  3 x 2 y 2  xy 3  x 3 y  3 x 2 y 2  3 xy 3  y 4 
        x 4  4 x 3 y  6 x 2 y 2  4 xy 3  y 4
               4 4  4 3       4 2 2  4 3  4 4
   ( x  y)    x    x y    x y    xy    y
           4
               0        1     2      3      4
                                            
                 n
                     n  i n i
   ( x  y)    x y
           n
                    i
               i 0  
CMSC 250                                                                              32

				
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