# CMSC 250 by yurtgc548

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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)

CMSC 250                                                                    2
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

CMSC 250                                                                      3
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 )

CMSC 250                                                                4
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)}

CMSC 250                                                   5
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

CMSC 250                                                      6
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?

CMSC 250                                                        7
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

CMSC 250                         9
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

CMSC 250                                                        10
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)

CMSC 250                                                        11
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)

CMSC 250                                                          12
PINs with less specified length-

   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

CMSC 250                                                              13

   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)

CMSC 250                                                          14
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

CMSC 250                         16
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
CMSC 250                                                   22
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.

CMSC 250                                                             23
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                             

CMSC 250                                                                   24
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)

CMSC 250                                                     25
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?

CMSC 250                                                                26
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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