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M.B.A 2 All right are reserved with CU-IDOL QUANTITATIVE TECHNIQUES FOR MANAGERS Course Code: MBA602 Semester : First SLM Unit : E-Lesson Unit: 7 5 www.cuidol.in Unit-7 (MBA602)

QUANTITATIVE TECHNIQUES FOR 33 MANAGERS OBJECTIVES INTRODUCTION Student will be able to : Probability theory is the branch of Articulate the axioms (laws) of probability mathematics concerned with probability. ... Typically these axioms formalise probability Distinguish between an event having zero in terms of a probability space, which assigns probability and an event being a measure taking values between 0 and 1, impossible termed the probability measure, to a set of outcomes called the sample space. Describe situations in which humans are bad at probabilistic reasoning Probabilities quantify uncertainty regarding the occurrence of events Define and identify some basic probability distributions/random variables Probability Theory gives rise to many interesting and important philosophical questions www.cuidol.in Unit-7 (MBA602) INASllTITriUgThEt aOrFeDreISsTeArNveCdE AwNitDh OCNUL-IIDNOE LLEARNING

TOPICS TO BE COVERED 4 > Probability Theory > Structure of Probability > Sample Space > Independent Events > Dependent Events > Probability Theory > Multiplication Law of Probability > Bayes Theorem www.cuidol.in Unit-7 (MBA602) All right are reserved with CU-IDOL

Probability Theory 5 • Probability is simply how likely something is to happen. • Probability means possibility. • It is a branch of mathematics that deals with the occurrence of a random event. • The value is expressed between zero and one. • Probability has been introduced in Maths to predict how likely events are to happen. • The meaning of probability is basically the extent to which something is likely to happen. This is the basic probability theory which is also used in the probability distribution, where we will learn about the possibility of outcomes for a random experiment. To find the probability of a single event to occur, first we should know the total number of possible outcomes. • Probability of an event happening = Number of ways it can happen/Total number of outcomes www.cuidol.in Unit-7 (MBA602) All right are reserved with CU-IDOL

Structure of Probability 6 • Experiment: a repeatable procedure with a set of possible results. Example: Throwing dice We can throw the dice again and again, so it is repeatable. The set of possible results from any single throw is {1, 2, 3, 4, 5, 6} • Outcome: A possible result of an experiment. Example: Getting a \"6\" www.cuidol.in Unit-7 (MBA602) All right are reserved with CU-IDOL

Sample Space 7 • Sample Space: all the possible outcomes of an experiment. Example: choosing a card from a deck • There are 52 cards in a deck (excluding Jokers) • So the Sample Space is all 52 possible cards: {Ace of Hearts, 2 of Hearts, etc... } www.cuidol.in Unit-7 (MBA602) All right are reserved with CU-IDOL

Events When we say \"Event\" we mean one (or more) 8 outcomes All right are reserved with CU-IDOL Example Events: • Getting a Tail when tossing a coin is an event • Rolling a \"5\" is an event. Events can be: • Independent (each event is not affected by other events), • Dependent (also called \"Conditional\", where an event is affected by other events) • Mutually Exclusive (events can't happen at the same time) www.cuidol.in Unit-7 (MBA602)

Independent Events 9 Events can be \"Independent\", meaning each event is not affected by any other events. This is an important idea! A coin does not \"know\" that it came up heads before ... each toss of a coin is a perfect isolated thing. Example: You toss a coin three times and it comes up \"Heads\" each time ... what is the chance that the next toss will also be a \"Head\"? • The chance is simply 1/2, or 50%, just like ANY OTHER toss of the coin. • What it did in the past will not affect the current toss! www.cuidol.in Unit-7 (MBA602) All right are reserved with CU-IDOL

Dependent Events 10 But some events can be \"dependent\" ... which means they can be affected by previous events. Example: Drawing 2 Cards from a Deck • After taking one card from the deck there are less cards available, so the probabilities change! • Let's look at the chances of getting a King. For the 1st card the chance of drawing a King is 4 out of 52. But for the 2nd card: • If the 1st card was a King, then the 2nd card is less likely to be a King, as only 3 of the 51 cards left are Kings. • If the 1st card was not a King, then the 2nd card is slightly more likely to be a King, as 4 of the 51 cards left are King. This is because we are removing cards from the deck. Replacement: When we put each card back after drawing it the chances don't change, as the events are independent. Without Replacement: The chances will change, and the events are dependent. www.cuidol.in Unit-7 (MBA602) All right are reserved with CU-IDOL

Mutually Exclusive 11 Mutually Exclusive means we can't get both events at the same time It is either one or the other, but not both Examples: • Turning left or right are Mutually Exclusive (you can't do both at the same time) • Heads and Tails are Mutually Exclusive • Kings and Aces are Mutually Exclusive www.cuidol.in Unit-7 (MBA602) All right are reserved with CU-IDOL

Example 12 • Kings and Hearts are not Mutually Exclusive, because we can have a King of Hearts! • Like here: Aces and Kings are Mutually Exclusive (can't be both) Hearts and Kings are Hearts and Kings are not Mutually Exclusive not Mutually Exclusive (can be both) (can be both) www.cuidol.in Unit-7 (MBA602) All right are reserved with CU-IDOL

Addition Laws of Probability 13 • These mutually exclusive events follow the addition law of probability • If the no. of mutually exclusive events are n and p1 is the individual probability • Total prob. P = p1+ p2 + ………….+ pn = 1 • P(A or B) = P(A) + P(B) www.cuidol.in Unit-7 (MBA602) All right are reserved with CU-IDOL

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Multiplication Law of Probability 15 • It is applied when two or more events are occurring together but they are independent of each other • If the no. of events are n and p1 is the individual probability • Total prob. P = p1 * p2 * ………….* pn • P(A and B) = P(A) * P(B) www.cuidol.in Unit-7 (MBA602) All right are reserved with CU-IDOL

Multiplication Law of Probability 16 • If we assume twin pregnancy will occur once in 80 preg. And child with Rh -ve blood group will be born once in 10 births • prob. of male child,p1 = ½ prob. of child being Rh +ve, p2 = 9/10 prob. of single birth, p3 = 79/80 • prob. of male, Rh +ve and single birth = ½*9/10*79/80 = 711/1600 www.cuidol.in Unit-7 (MBA602) All right are reserved with CU-IDOL

Conditional Probability 17 • Conditional probability is the probability that an event will occur given that another has already occurred. • If A and B are two events, then prob. Of A given that B has already occurred will be : P(A|B) = P(A and B) / P(B) P(B|A) = P(A and B) / P(A) www.cuidol.in Unit-7 (MBA602) All right are reserved with CU-IDOL

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Bayes’ Theorem 21 • The concept of conditional probability can be revised based on new information and to determine the probability that a particular effect was due to specific causes. The procedures for revising these probabilities is known as Bayes’ theorem. www.cuidol.in Unit-7 (MBA602) All right are reserved with CU-IDOL

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SUMMARY 25  ProbabilityTheory  Additional Law of Probability  Structure of Probability  Multiplication Law of Probability  Sample Space  Conditional Probability  Events  Bayes Theorem  Independent Events  Dependent Events Unit-7 (MBA602) All right are reserved with CU-IDOL  Mutually Exclusive www.cuidol.in

MULTIPLE CHOICE QUESTIONS 1) Mutually exclusive events 26 A) Contain all sample points C) Contain common sample points B) Does not contain any sample points D) Does not contain any common sample points 2) What is the probability of an impossible event? A) 0 C) 1 B) Not defined D) Insufficient data 3) What are the chances that 2 boys are sitting together for a photograh if there are 5 girls and 2 boys ? A) 1/21 C) 4/7 B) 2/7 D) 5/7 4) Let A and B be 2 events such that the occurence of Aimplies occurence of B, but not vice-versa. Then the correct relation between P(a) and P (b) is............... A) P(A) < P(B) C) P(B) ≥ P(A) Ans. 1. (d) 2. (a) 3. (d) 4. B) P(A) = P(B) D) P(A) ≥ P(B) All right are reserved with CU-IDOL (b) www.cuidol.in Unit-7 (MBA602)

FREQUENTLY ASKED QUESTIONS 27 Q1. What do you understand by probability? Ans. Probability means possibility. - It is a branch of mathematics that deals with the occurrence of a random event. - The value is expressed between zero and one. - Probability has been introduced in Maths to predict how likely events are to happen For further details Refer to the SLM. Q2. Explain 'Conditional Probability'? Conditional probability is the probability that an event will occur given that another has already occurred. For further details Refer to the SLM. Q3. What are the various forms of 'Events'? Ans. Events can be: Independent (each event is not affected by other events), Dependent (also called \"Conditional\", where an event is affected by other events) Mutually Exclusive (events can't happen at the same time) For further details Refer to the SLM. www.cuidol.in Unit-7 (MBA602) All right are reserved with CU-IDOL

REFERENCES 28  Stigler, Cambridge M.A., “The History of Statistics”, 1986, Belknap Press  Grant, L.E. and R.C. Leavenworth, “Statistical Quality Control” 1996 McGraw Hill - Book Co.  Tufte, E.R.,”The Visual display of Quantitative Information”, 1983, Graphics Press.  Rowntree, D., “Probability” 1984, Charles Scribner's Sons.  Levin, R.I., and D.S. Rubin, “Statistics for Management”, 1997, Prentice Hall (India)  Gupta, S.C., “Fundamentals of Statistics”, Himalaya Publishing House, Mumbai. www.cuidol.in Unit-7 (MBA602) All right are reserved with CU-IDOL

29 THANK YOU For queries Email: [email protected] www.cuidol.in Unit-7 (MBA602) All right are reserved with CU-IDOL 