PROBABILITY & STATISTICS

1 MCASEMESTER-IIPROBABILITY &STATISTICSmca-52301 PROBABILITYINTRODUCTION TO PROBABILITYM anagers need to cope with uncertainty in many decisionmaking situations. For example, you as a manager may assumethat the volume of sales in the successive year is known exactly toyou. This is not true because you know roughly what the next yearsales will be. But you cannot give the exact number. There is someuncertainty. Concepts of PROBABILITY will help you to measureuncertainty and perform associated analyses. This unit provides theconceptual framework of PROBABILITY and the various probabilityrules that are essential in business 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 onthe problem situation.

2 Make use of the expected values for and SubsetsThe lesson introduces the important topic of sets, a simpleidea that recurs throughout the study of PROBABILITY and Definitions Asetis a well-defined collection of objects. Each object in a set is called anelementof the set. Two sets areequalif they have exactly the same elementsin them. A set that contains no elements is called anull setor anemptyset. If every element in SetAis also in SetB, then SetAis asubsetof Notation A set is usually denoted by a capital letter, such asA, B,orC. An element of a set is usually denoted by a smallletter, suchasx, y,orz.

3 A set may be decribed by listing all of its elements enclosedin braces. For example, if SetAconsists of the numbers 2,4, 6, and 8, we may say:A= {2, 4, 6, 8}. The null set is denoted by { }.mca-5231 Sets may also be described by stating a rule. We coulddescribe SetAfrom the previous example by stating: SetAconsists of all the even single-digit positive OperationsSuppose we have four sets-W, X, Y, and Z. Let these sets bedefined as follows: W = {2}; X = {1, 2}; Y= {2, 3, 4}; and Z = {1, 2, 3,4}. Theunionof two sets is the set of elements that belong toone or both of the two sets.

4 Thus, set Z is the union of sets Xand Y. Symbolically, the union of X and Y is denoted by X Y. Theintersectionof two sets is the set of elements that arecommon to both sets. Thus, set W is the intersection of setsX and Y. Symbolically, the intersection of X and Y is denoted by X the set of A isthe set of vowels, then A could be described asA= {a,e, i, o, u}. the set of positive it would be impossible to listallof the positiveintegers, we need to use a rule to describe this set. Wemight sayAconsists of all integers greater than {1, 2, 3} and SetB= {3, 2, 1}. Is SetAequal to SetB?

5 Yes. Two sets are equal if they have the same order in which the elements are listed does not is the set of men with four arms?Since all men have two arms at most, the set of men withfour arms contains no elements. It is the null set (or emptyset). {1, 2, 3} and SetB= {1, 2, 4, 5, 6}. Is SetAa subsetof SetB?SetAwould be a subset of SetBif every element from SetAwere also in SetB. However, this is not the case. Thenumber 3 is in SetA, but not in SetB. Therefore, SetAis nota subset of ExperimentsAllstatistical experimentshave three things in common: The experiment can have more than one possible outcome.

6 Each possible outcome can be specified in The outcome of the experiment depends on coin toss has all the attributes of a statistical experiment. There ismore than one possible outcome. We can specify each possibleoutcome ( , heads or tails) in advance. And there is an element ofchance, since the outcome is Sample Space Asample spaceis a set of elements that represents allpossible outcomes of a statistical experiment. Asample pointis an element of a sample space. Aneventis a subset of a sample space-one or moresample points. Types of events Two events aremutually exclusiveif they have no samplepoints in common.

7 Two events areindependentwhen the occurrence of onedoes not affect the PROBABILITY of the occurrence of the I roll a die. Is that a statistical experiment?Yes. Like a coin toss, rolling dice is a statistical is more than one possible outcome. We can specify eachpossible outcome in advance. And there is an element you roll a single die, what is the sample sample space is all of the possible outcomes-an integerbetween 1 and of the following are sample points when you roll a die-3,6,and9?The numbers 3 and 6 are sample points, because they are inthe sample space.

8 The number 9 is not a sample point, since itis outside the sample space; with one die, the largest numberthat you can roll is ofthe following sets represent an event when you roll adie?A.{1}B.{2, 4,}C.{2, 4, 6} of the aboveThe correct answer is D. Remember that an event is a subset ofa sample space. The sample space is any integer from 1 to of the sets shown above is a subset of the sample space,so each represents an 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 incommon. Events A and B are mutually exclusive, and Events A andC are mutually exclusive; since they have no points in B and C have common sample points, so they are notmutually you roll a die two times.

9 Is each roll ofthe die anindependentevent?Yes. Two events are independent when the occurrence of onehas no effect on the PROBABILITY of the occurrence of the roll of the die affects the outcome of the other roll; soeach roll of the die is ProbabilityTheprobabilityof asample pointis a measure of the likelihoodthat the sample point will of a Sample PointBy convention, statisticianshave 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 asamplespaceis equal to 1.

10 Example 1 Suppose we conduct a simplestatistical experiment. We flip a coinone time. The coin flip can have one of two outcomes-heads ortails. Together, these outcomes represent the sample space of ourexperiment. Individually, each outcome represents a sample pointin the sample space. What is the PROBABILITY of each sample point?Solution:The sum of probabilities of all the sample points mustequal 1. And the PROBABILITY of getting a head is equal to theprobability of getting a tail. Therefore, the PROBABILITY of eachsample point (heads or tails) must be equal to 1 2 Let's repeat the experiment of Example 1, with a die instead of acoin.