Counting Techniques & Probability: Situations, Sample Space, Events, & Calculation, Slides of Discrete Mathematics

Download Counting Techniques & Probability: Situations, Sample Space, Events, & Calculation and more Slides Discrete Mathematics in PDF only on Docsity!

1

Counting

2

Situations where counting

techniques are used

You toss a pair of dice in a casino game.

You win if the numbers showing face up

have a sum of 7.

Question: What are your chances of

winning the game?

4

Situations where counting

techniques are used

 There are 4 jobs that should be processed

on the same machine. ( Can’t be processed simultaneously ).

Here is an example of a possible schedule:

 Question: What is the number of all possible

schedules?

Job 3 Job 1 Job 4 Job 2

5

Situations where counting

techniques are used

 Consider the following nested loop:

for i:=1 to 5 for j:=1 to 6 [ Statement 1 ; Statement 2. ] next j next i

 Question: How many times the statements in the

inner loop will be executed?

7

Random Processes,

Sample Space and Events

 A proces s is called random if

• a set of different outcomes are possible;
• one of the outcomes is sure to occur;
• but it is impossible to predict with certainty which outcome that will be.

 A sample space is the set of all possible outcomes

of a random process or experiment.

 An event is a subset of a sample space.

8

Probability

 If S is a finite sample space

(in which all outcomes are equally likely), E is an event in S,

then the probability of E is

 Notation: For any finite set A,

n(A) denotes the number of elements in A.

Then

the totalnumber of outcomesin S

the number of outcomesin E

P ( E ) =

( )

( ) ( ) n S

n E P E =

Applying the dice example in Monopoly Game

• Your opponent’s token is in one of the squares
• His turn consists of rolling two dice and moving the token clockwise on the board the number of squares indicated by the sum of dice values
• When his token lands on a property

that is owned by you, you collect rent

• It is more advantageous to have houses or

hotels on your properties because rents are much higher than for unimproved properties

• You might build houses or hotels on

your properties before your opponent rolls the dice

• Suppose you own most of the squares following (clockwise) your opponent’s token. In which square should you build houses or hotels?

11

Number of Elements in a List

 If m and n are integers and m ≤ n ,

then there are n-m+1 integers

from m to n inclusive.

 Example:

a) How many elements are there in the array

A[12], A[13], …, A[75], A[76]?

b) What is the probability

that a randomly chosen element of the array

has a subscript which is divisible by 7?