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**Biju Patnaik University of Technology BPUT - BPUT**- Master of Computer Applications
- MCA
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mca-5 230 1 PROBABILITY INTRODUCTION TO PROBABILITY Managers need to cope with uncertainty in many decision making situations. For example, you as a manager may assume that the volume of sales in the successive year is known exactly to you. This is not true because you know roughly what the next year sales will be. But you cannot give the exact number. There is some uncertainty. Concepts of probability will help you to measure uncertainty and perform associated analyses. This unit provides the conceptual framework of probability and the various probability rules that are essential in business decisions. Learning objectives: After reading this unit, you will be able to: Appreciate the use of probability in decision making Explain the types of probability Define and use the various rules of probability depending on the problem situation. Make use of the expected values for decision-making. Probability Sets and Subsets The lesson introduces the important topic of sets, a simple idea that recurs throughout the study of probability and statistics. Set Definitions A set is a well-defined collection of objects. Each object in a set is called an element of the set. Two sets are equal if they have exactly the same elements in them. A set that contains no elements is called a null set or an empty set. If every element in Set A is also in Set B, then Set A is a subset of Set B. Set Notation A set is usually denoted by a capital letter, such as A, B, or C. An element of a set is usually denoted by a small letter, such as x, y, or z. A set may be decribed by listing all of its elements enclosed in braces. For example, if Set A consists of the numbers 2, 4, 6, and 8, we may say: A = {2, 4, 6, 8}. The null set is denoted by {∅}.

mca-5 231 Sets may also be described by stating a rule. We could describe Set A from the previous example by stating: Set A consists of all the even single-digit positive integers. Set Operations Suppose we have four sets - W, X, Y, and Z. Let these sets be defined as follows: W = {2}; X = {1, 2}; Y= {2, 3, 4}; and Z = {1, 2, 3, 4}. The union of two sets is the set of elements that belong to one or both of the two sets. Thus, set Z is the union of sets X and Y. Symbolically, the union of X and Y is denoted by X ∪ Y. The intersection of two sets is the set of elements that are common to both sets. Thus, set W is the intersection of sets X and Y. Symbolically, the intersection of X and Y is denoted by X ∩ Y. Sample Problems 1. Describe the set of vowels. If A is the set of vowels, then A could be described as A = {a, e, i, o, u}. 2. Describe the set of positive integers. Since it would be impossible to list all of the positive integers, we need to use a rule to describe this set. We might say A consists of all integers greater than zero. 3. Set A = {1, 2, 3} and Set B = {3, 2, 1}. Is Set A equal to Set B? Yes. Two sets are equal if they have the same elements. The order in which the elements are listed does not matter. 4. What is the set of men with four arms? Since all men have two arms at most, the set of men with four arms contains no elements. It is the null set (or empty set). 5. Set A = {1, 2, 3} and Set B = {1, 2, 4, 5, 6}. Is Set A a subset of Set B? Set A would be a subset of Set B if every element from Set A were also in Set B. However, this is not the case. The number 3 is in Set A, but not in Set B. Therefore, Set A is not a subset of Set B. Statistical Experiments All statistical experiments have three things in common: The experiment can have more than one possible outcome. Each possible outcome can be specified in advance.

mca-5 232 The outcome of the experiment depends on chance. A coin toss has all the attributes of a statistical experiment. There is more than one possible outcome. We can specify each possible outcome (i.e., heads or tails) in advance. And there is an element of chance, since the outcome is uncertain. The Sample Space A sample space is a set of elements that represents all possible outcomes of a statistical experiment. A sample point is an element of a sample space. An event is a subset of a sample space - one or more sample points. Types of events Two events are mutually exclusive if they have no sample points in common. Two events are independent when the occurrence of one does not affect the probability of the occurrence of the other. Sample Problems 1. Suppose I roll a die. Is that a statistical experiment? Yes. Like a coin toss, rolling dice is a statistical experiment. There is more than one possible outcome. We can specify each possible outcome in advance. And there is an element of chance. 2. When you roll a single die, what is the sample space. The sample space is all of the possible outcomes - an integer between 1 and 6. 3. Which of the following are sample points when you roll a die - 3, 6, and 9? The numbers 3 and 6 are sample points, because they are in the sample space. The number 9 is not a sample point, since it is outside the sample space; with one die, the largest number that you can roll is 6. 4. Which of the following sets represent an event when you roll a die? A. B. C. D. {1} {2, 4,} {2, 4, 6} All of the above The correct answer is D. Remember that an event is a subset of a sample space. The sample space is any integer from 1 to 6.

mca-5 233 Each of the sets shown above is a subset of the sample space, so each represents an event. 5. Consider the events listed below. Which are mutually exclusive? A. {1} B. {2, 4,} C. {2, 4, 6} Two events are mutually exclusive, if they have no sample points in common. Events A and B are mutually exclusive, and Events A and C are mutually exclusive; since they have no points in common. Events B and C have common sample points, so they are not mutually exclusive. 6. Suppose you roll a die two times. Is each roll of the die an independent event? Yes. Two events are independent when the occurrence of one has no effect on the probability of the occurrence of the other. Neither roll of the die affects the outcome of the other roll; so each roll of the die is independent. Basic Probability The probability of a sample point is a measure of the likelihood that the sample point will occur. Probability of a Sample Point By convention, statisticians have agreed on the following rules. The probability of any sample point can range from 0 to 1. The sum of probabilities of all sample points in a sample space is equal to 1. Example 1 Suppose we conduct a simple statistical experiment. We flip a coin one time. The coin flip can have one of two outcomes - heads or tails. Together, these outcomes represent the sample space of our experiment. Individually, each outcome represents a sample point in the sample space. What is the probability of each sample point? Solution: The sum of probabilities of all the sample points must equal 1. And the probability of getting a head is equal to the probability of getting a tail. Therefore, the probability of each sample point (heads or tails) must be equal to 1/2. Example 2 Let's repeat the experiment of Example 1, with a die instead of a coin. If we toss a fair die, what is the probability of each sample point? Solution: For this experiment, the sample space consists of six sample points: {1, 2, 3, 4, 5, 6}. Each sample point has equal probability. And the sum of probabilities of all the sample points

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