Discrete Structures Chapter 4

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Discrete StructuresChapter 4 Counting and
Probability
Nurul Amelina Nasharuddin Multimedia Department
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Outline

• Rules of Sum and Product
• Permutations
• Combinations The Binomial Theorem
• Combinations with Repetition Distribution
• Probability

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Combinations with Repetition

• Example
• How many ways are there to select 4 pieces of
  fruits from a bowl containing apples, oranges,
  and pears if the order does not matter, only the
  type of fruit matters, and there are at least 4
  pieces of each type of fruit in the bowl

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Answer

• Some of the results

Possible Selection Representation
A A A O X X X X
A A A A X X X X
A A O P X X X X
P P P P X X X X
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Answer

• The number of ways to select 4 pieces of fruit
  The number of ways to arrange 4 Xs and 2 s,
  which is given by
• 6! / 4!(6-4)! C(6,4) 15
  ways.

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Combinations with Repetition

• In general, when we wish to select, with
  repetition, r of n distinct elements, we are
  considering all arrangements of r Xs and n-1 s
  and that their number is

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Combinations with Repetition

• An r-combination of a set of n elements is an
  unordered selection of r elements from the set,
  with repetition is

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Example (1)

• A person throwing a party wants to set out 15
  assorted cans of drinks for his guests. He shops
  at a store that sells five different types of
  soft drinks. How many different selections of 15
  cans can he make?
• (Here n 5, r 15)

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Example (1)

• 4 s (to separate the categories of soft drinks)
• 15 Xs (to represent the cans selected)
• 19! / 15!(19-15)!
• C(19,15)
• 3876 ways.

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Example (2)

• A donut shop offers 20 kinds of donuts. Assuming
  that there are at least a dozen of each kind when
  we enter the shop, we can select a dozen donuts
  in
• (Here n 20, r 12).
• C(31, 12) 141,120,525
  ways.

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Example (3)

• A restaurant offers 4 kinds of food. In how many
  ways can we choose six of the food?
•  
• C(6 4 - 1, 6) C(9, 6)
• C(9,
  3)
• 9!
  84 ways.
• 3!
  6!

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Which formula to use?
Different ways of choosing k elements from n
Order Relevant Order Does Not Relevant
Repetition is Allowed nk
Repetition is Not Allowed P(n, k)
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Counting and Probability
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Discrete Probability

• The probability of an event is the likelihood
  that event will occur.
• Probability 1 means that it must happen while
  probability 0 means that it cannot happen
• Eg The probability of
• Manchester United defeat Liverpool this season
  is 1
• Liverpool win the Premier League this season is
  0
• Events which may or may not occur are assigned a
  number between 0 and 1.

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Discrete Probability

• Consider the following problems
• Whats the probability of tossing a coin 3 times
  and getting all heads or all tails?
• Whats the probability that a list consisting of
  n distinct numbers will not be sorted?

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Discrete Probability

• An experiment is a process that yields an outcome
• A sample space is the set of all possible
  outcomes of a random process
• An event is an outcome or combination of outcomes
  from an experiment
• An event is a subset of a sample space
• Examples of experiments
• - Rolling a six-sided die
• - Tossing a coin

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Example

• Experiment 1 Tossing a coin.
• Sample space S Head or Tail or we could
  write S 0, 1 where 0 represents a tail and 1
  represents a head.
• Experiment 2 Tossing a coin twice
• S HH, TT, HT, TH where some events
• E1 Head,
• E2 Tail,
• E3 All heads

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Definition of Probability

• Suppose an event E can happen in r ways out of a
  total of n possible equally likely ways.
• Then the probability of occurrence of the event
  (called its success) is denoted by
• The probability of non-occurrence of the event
  (called its failure) is denoted by
• Thus,

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Definition of Probabilityusing Sample Spaces

• If S is a finite sample space in which all
  outcomes are equally likely and E is an event in
  S, then the probability of E, P(E), is
• where
• N(E) is the number of outcomes in E
• N(S) is the total number of outcomes in S

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Example (1)

• Whats the probability of tossing a coin 3 times
  and getting all heads or all tails?
• Can consider set of ways of tossing coin 3 times
  Sample space, S HHH, HHT, HTH, HTT, THH, THT,
  TTH, TTT
• Next, consider set of ways of tossing all heads
  or all tails Event, E HHH, TTT
• Assuming all outcomes equally likely to occur
• ? P(E) 2/8 0.25

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Example (2)

• Five microprocessors are randomly selected from a
  lot of 1000 microprocessors among which 20 are
  defective. Find the probability of obtaining no
  defective microprocessors.
• There are C(1000,5) ways to select 5 micros.
• There are C(980,5) ways to select 5 good micros.
• The prob. of obtaining no defective micros is
• C(980,5)/C(1000,5) 0.904

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Probability of Combinations of Events

• Theorem Let E1 and E2 be events in the sample
  space S. Then
• P(E1 ? E2) P(E1) P(E2) P(E1 ? E2)
• Eg What is the probability that a positive
  integer selected at random from the set of
  positive integers not greater than 100 is
  divisible by either 2 or 5
• E1 Event that the integer selected is
  divisible by 2
• E2 Event that the integer selected is
  divisible by 5
• P(E1 ? E2) 50/100 20/100 10/100
• 3/5

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Exercise

1. If any seven digits could be used to form a
   telephone number, how many seven-digits telephone
   numbers would not have repeated digits?
2. How many seven-digit telephone numbers would have
   at least one repeated digit?
3. What is the probability that a randomly chosen
   seven-digit telephone number would have at least
   one repeated digit?

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Answer

1. 10 x 9 x 8 x 7 x 6 x 5 x 4 604800
2. no of PN with at least one digit
   total no of PN no of PN with no
   repeated digit
   107 604800 9395200
3. 9395200 / 107 0.93952

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Counting Elements of Sets

• The Principle of Inclusion/Exclusion Rule for Two
  or Three Sets
• If A, B, and C are finite sets, then
• N(A?B) N(A) N(B) N(A ? B)
• and
• N(A?B?C) N(A) N(B) N(C) N(A?B)
  N(A?C) N(B?C) N(A?B?C)

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Example (1)

• In a class of 50 college freshmen, 30 are
  studying BASIC, 25 studying PASCAL, and 10 are
  studying both.
• How many freshmen are not studying either
  computer language?
• A set of freshmen study BASIC
• B set of freshmen study PASCAL
• N(A?B) N(A)N(B)-N(A?B)
• 30 25 10 45
• ?Not studying either 50 45 5

10
20 10
15
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Example (2)

• A professor takes a survey to determine how many
  students know certain computer languages. The
  finding is that out of a total of 50 students in
  the class,
• 30 know Java
• 18 know C
• 26 know SQL
• 9 know both Java and C
• 16 know both Java and SQL
• 8 know both C and SQL
• 47 know at least one of the 3 languages.

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Example (2)

• a. How many students know none of the three
  languages?
• b. How many students know all three languages?
• c. How many students know Java and C but not
  SQL? How many students know Java but neither C
  nor SQL
• Answer
• 50 47 3
• ?
• ?

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Example (2)

• J the set of students who know Java
• C the set of students who know C
• S the set of students who know SQL
• Use Inclusion/Exclusion rule.