Nor Fashihah Mohd Noor Institut Matematik Kejuruteraan Universiti Malaysia Perlis ІМ ќ INSTITUT MATEMATIK K E J U R U T E R A A N U N I M A P

 When you toss a single coin, you will see either a head (H) or a tail (T). If you toss the coin repeatedly, you will generate an infinitely large number of Hs and Ts – the entire population. What does the population look like? If the coin is fair, then the population should contain 50% Hs and 50% Ts. Now toss the coin one more time. What is the chance to getting a head? Most people would say that the “ probability ” or chance is ½. 3

Probability  Probability is a measure of the likelihood of an event A occurring in one experiment or trial and it is denoted by P (A). Experiment  An experiment is any process of making an observation leading to outcomes for a sample space. 4

The mathematical basis of probability is the theory of sets. Definition 1.2: Sample Spaces, Sets and Events  Sets A set is a collection of elements or components  Sample Spaces, S A sample space consists of points that correspond to all possible outcomes.  Events An event is a set of outcomes of an experiment and a subset of the sample space. 5

Let A denote the events of obtaining a number which could divide by 3 in an experiment of tossing a dice, Hence, A = {3, 6} is a subset of S = {1,2,3,4,5,6} Basic Operations Figure 1.1: Venn diagram representation of events 6 C B A S

1. The union of events A and B, which is denoted as, is the set of all elements that belong to A or B or both. Two or more events are called collective exhaustive events if the unions of these events result in the sample space. 2. The intersection of events A and B, which is denoted by, is the set of all elements that belong to both A and B. When A and B have no outcomes in common, they are said to be mutually exclusive or disjoint sets. 3. The event that contains all of the elements that do not belong to an event A is called the complement of A and is denoted by 7

Given the following sets; A = {2, 4, 6, 8, 10} B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} C = {1, 3, 5, 11, ……}, the set of odd numbers D = {failure of a structure due to an earth quake} E = {failure of a structure due to strong winds} 8

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Example 1.3 In a carom game, there are 3 white seeds and 2 black seeds. Three seeds have been successfully converted in the game. Assuming that the convert is being done one after another and the rules of carom game are disobeyed, calculate the probability.

Solution Let, M = white seed and K = black seed 11

This counting rule allows one to count the number of outcomes when the experiment involves selecting r objects from a set of n objects.

Example 1.4 Suppose that in the taste test, each participant samples eight products and is asked the three best products, but not in any particular order. Solution The number of possible answer test is then

This counting rule allows one to compute the number of outcomes when r objects are to be selected from a set of n objects where the order of selection is important.

Example 1.5 Three lottery tickets are drawn from a total of 50. If the tickets will be distributed to each of the three employees in the order in which they are drawn, the order will be important. How many simple events are associated with the experiment? Solution The total number of simple events is

Probability Axioms: Axiom 1 : For any event, A, P(A) ≥ 0 Axiom 2 : P(S) = 1 Axiom 3 : For any countable collection A 1, A 2, ……… of mutually exclusive events Theorem 1.1 : Laws of Probability

Example 1.6 Two fair dice are thrown. Determine a) the sample space of the experiment b) the elements of event A if the outcomes of both dice thrown are showing the same digit. c) the elements of event B if the first thrown giving a greater digit than the second thrown. d) probability of event A, P(A) and event B, P(B)

(1, 1)(1, 2)(1, 3)(1, 2)(1, 5)(1, 6) 2(2, 1)(2, 2)(2, 3)(2, 4)(2, 5)(2, 6) 3(3, 1)(3, 2)(3, 3)(3, 4)(3, 5)(3, 6) 4(4, 1)(4, 2)(4, 3)(4, 4)(4, 5)(4, 6) 5(5, 1)(5, 2)(5, 3)(5, 4)(5, 5)(5, 6) 6(6, 1)(6, 2)(6, 3)(6, 4)(6, 5)(6, 6)

b) A = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} c) B = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3), (5, 4), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5)}

Example 1.7 Consider randomly selecting a KUKUM Master Degree international student, and let A denote the event that the selected individual has a Visa Card and B has a Master Card. Suppose that P(A) = 0.5 and P(B) = 0.4 and = a) Compute the probability that the selected individual has at least one of the two types of cards ? b) What is the probability that the selected individual has neither type of card?

Example 1.8 In an assessment for students who took Engineering Mathematics I course, it is known that the percentage of students who passed in their monthly test is 80% while 85% passed their quiz and 75% passed in both monthly test and quiz. A students is selected at random, calculate the probability that a) passed the monthly test or quiz b) passed the monthly test but failed quiz c) failed both monthly test and quiz

Solutions Let, U = student passed the monthly test K = student passed the quiz

Example 1.9 In a certain residential area, 60% of all households subscribe to Berita Harian newspaper, 80% subscribe to The Star paper, and 50% of all households subscribe to both papers. If a household is selected at random, what is the probability that it subscribes a) at least one of the two newspaper b) exactly one of the two newspaper

Solutions Let, A = subscribes to Berita Harian paper B = subscribes to The Star paper

Example 1.10 A box in a certain supply room contains four 40-W lightbulbs, two 60-W bulbs and eight 75-W bulbs. One lightbulb is randomly selected, calculate the probability that the selected lightbulb is rated 40-W or 60-W ? Solution Sample space, S = {four 40-W lightbulbs, two 60-W bulbs, eight 75-W bulbs } Then, n(S) = 14.

Let the events A and B be, A = 40-W lightbulbs is selected B = 60-W lightbulbs is selected Hence, Events A and B are said to be mutually exclusive because of the two events couldn’t occur at the same time. Thus,

Definition 1.4 : For any two events A and B with P(B) > 0, the conditional probability of A given that B has occurred is defined by

Area/GenderMaleFemaleTotal Urban Rural Total If a student is selected at random and have been told that the individual is a male student, what is the probability of he is from urban area?

Probability of a person is from urban area and it is known that the individual is a male student,

Transmission type/Colour GreyBlueBlackRed Automatic Manual

Solution

Definition 1.5 : Two events A and B are said to be independent if and only if either Otherwise, the events are said to be dependent.