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Discrete Structures Chapter 4 Counting and Probability. Nurul Amelina Nasharuddin Multimedia Department. Outline. Rules of Sum and Product Permutations Combinations: The Binomial Theorem Combinations with Repetition: Distribution Probability. Combinations with Repetition. Example:

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Discrete Structures Chapter 4 Counting and Probability

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Discrete structures chapter 4 counting and probability

Discrete StructuresChapter 4

Counting and Probability

Nurul Amelina Nasharuddin

Multimedia Department


Outline

Outline

  • Rules of Sum and Product

  • Permutations

  • Combinations: The Binomial Theorem

  • Combinations with Repetition: Distribution

  • Probability


Combinations with repetition

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


Answer

Answer

  • Some of the results:


Answer1

Answer

The number of ways to select 4 pieces of fruit = The number of ways to arrange 4 X’s and 2 |’s, which is given by

= 6! / 4!(6-4)! = C(6,4) = 15 ways.


Combinations with repetition1

Combinations with Repetition

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


Combinations with repetition2

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:


Example 1

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)


Example 11

Example (1)

  • 4 |’s (to separate the categories of soft drinks)

  • 15 X’s (to represent the cans selected)

    = 19! / 15!(19-15)!

    = C(19,15)

    = 3876 ways.


Example 2

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.


Example 3

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!


Which formula to use

Which formula to use?

Different ways of choosing k elements from n


Counting and probability

Counting and Probability


Discrete probability

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.


Discrete probability1

Discrete Probability

Consider the following problems:

  • What’s the probability of tossing a coin 3 times and getting all heads or all tails?

  • What’s the probability that a list consisting of n distinct numbers will not be sorted?


Discrete probability2

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


Example

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}


Definition of probability

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,


Definition of probability using sample spaces

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


Example 12

Example (1)

What’s 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


Example 21

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


Probability of combinations of events

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


Exercise

Exercise

  • If any seven digits could be used to form a telephone number, how many seven-digits telephone numbers would not have repeated digits?

  • How many seven-digit telephone numbers would have at least one repeated digit?

  • What is the probability that a randomly chosen seven-digit telephone number would have at least one repeated digit?


Answer2

Answer

  • 10 x 9 x 8 x 7 x 6 x 5 x 4 = 604800

  • [no of PN with at least one digit] = [total no of PN] – [no of PN with no repeated digit] = 107 – 604800 = 9395200

  • 9395200 / 107 = 0.93952


Counting elements of sets

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)


Example 13

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


Example 22

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.


Example 23

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

  • ?

  • ?


Example 24

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.


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