Probability Theory PDF

Theory of Probability Probability Theory UNIT - I Probability Distributions 1 Contents 1. Introduction 2. Simple Definitions 3. Types of Probability 4. Theorems of Probability 5. Probabilities under conditions of...

Theory of Probability Probability Theory UNIT - I Probability Distributions 1 Contents 1. Introduction 2. Simple Definitions 3. Types of Probability 4. Theorems of Probability 5. Probabilities under conditions of statistically independent events 6. Probabilities under conditions of statistically dependent events 7. Bayes Theorem 8. Glossary of Terms 2 Introduction If an experiment is repeated under essentially homogeneous & similar conditions we generally come across 2 types of situations:  Deterministic/ Predictable: - The result of what is usually known as the ‘outcome’ is unique or certain. Example:- The velocity ‘v’ of a particle after time ‘t’ is given by v = u + at Equation uniquely determines v if the right- hand quantities are known. 3  Unpredictable/ Probabilistic: - The result is not unique but may be one of the several possible outcomes. Examples: - (i) In tossing of a coin one is not sure if a head or a tail will be obtained. (ii) If a light tube has lasted for t hours, nothing can be said about its further life. It may fail to function any moment. 4 Simple Definitions Trial & Event  Example: - Consider an experiment which, though repeated under essentially identical conditions, does not give unique results but may result in any one of the several possible outcomes.  Experiment is known as a Trial & the outcomes are known as Events or Cases. Throwing a die is a Trial & getting 1 (2,3,…,6) is an event. Tossing a coin is a Trial & getting Head (H) or Tail (T) is an event. 5  Exhaustive Events: - The total number of possible outcomes in any trial.  In tossing a coin there are 2 exhaustive cases, head & tail.  In throwing a die, there are 6 exhaustive cases since any one of the 6 faces 1,2,…,6 may come uppermost. Experiment Collectively Exhaustive Events In a tossing of an unbiased coin Possible solutions – Head/ Tail Exhaustive no. of cases – 2 In a throw of an unbiased cubic Possible solutions – 1,2,3,4,5,6 die Exhaustive no. of cases – 6 In drawing a card from a well Possible solutions – Ace to King shuffled standard pack of playing Exhaustive no. of cases – 52 cards 6  Favorable Events/ Cases: - It is the number of outcomes which entail the happening of an event.  In throwing of 2 dice, the number of cases favorable to getting the sum 5 is: (1,4), (4,1), (2,3), (3,2).  In drawing a card from a pack of cards the number of cases favorable to drawing an ace is 4, for drawing a spade is 13 & for drawing a red card is 26. Independent Events: - If the happening (or non- happening) of an event is not affected by the supplementary knowledge concerning the occurrence of any number of the remaining events.  In tossing an unbiased coin the event of getting a head in the first toss is independent of getting a head in the second, third & subsequent throws. 7  Mutually exclusive Events: - If the happening of any one of the event precludes the happening of all the others.  In tossing a coin the events head & tail are mutually exclusive.  In throwing a die all the 6 faces numbered 1 to 6 are mutually exclusive since if any one of these faces comes, the possibility of others, in the same trial, is ruled out. Experiment Mutually Exclusive Events In a tossing of an unbiased Head/ Tail coin In a throw of an unbiased Occurrence of 1 or 2 or 3 or 4 or 5 or cubic die 6 In drawing a card from a well Card is a spade or heart shuffled standard pack of Card is a diamond or club playing cards Card is a king or a queen 8  Equally likely Events: - Outcomes of a trial are said to be equally likely if taken into consideration all the relevant evidences, there is no reason to expect one in preference to the others.  In tossing an unbiased coin or uniform coin, head or tail are equally likely events.  In throwing an unbiased die, all the 6 faces are equally likely to come. Experiment Collectively Exhaustive Events In a tossing of an unbiased Head is likely to come up as a Tail coin In a throw of an unbiased Any number out of 1,2,3,4,5,6 is cubic die likely to come up In drawing a card from a well Any card out of 52 is likely to come shuffled standard pack of up playing cards 9  Probability: Probability of a given event is an expression of likelihood of occurrence of an event.  Probability is a number which ranges from 0 to 1.  Zero (0) for an event which cannot occur and 1 for an event which is certain to occur. Importance of the concept of Probability  Probability models can be used for making predictions.  Probability theory facilitates the construction of econometric model.  It facilitates the managerial decisions on planning and control. 10 Types of Probability There are 3 approaches to probability, namely: 1. The Classical or ‘a priori’ probability 2. The Statistical or Empirical probability 3. The Axiomatic probability 11 Mathematical/ Classical/ ‘a priori’ Probability Basic assumption of classical approach is that the outcomes of a random experiment are “equally likely”. According to Laplace, a French Mathematician: “Probability, is the ratio of the number of ‘favorable’ cases to the total number of equally likely cases”. If the probability of occurrence of A is denoted by p(A), then by this definition, we have: Number of favorable cases m p = P(E) = ------------------------------ = ---- Total number of equally likely cases n 12  Probability ‘p’ of the happening of an event is also known as probability of success & ‘q’ the non- happening of the event as the probability of failure. If P(E) = 1, E is called a certain event & if P(E) = 0, E is called an impossible event The probability of an event E is a number such that 0 ≤ P(E) ≤ 1, & the sum of the probability that an event will occur & an event will not occur is equal to 1. i.e., p + q = 1 13 Limitations of Classical definition Classical probability is often called a priori probability because if one keeps using orderly examples of unbiased dice, fair coin, etc. one can state the answer in advance (a priori) without rolling a dice, tossing a coin etc. Classical definition of probability is not very satisfactory because of the following reasons:  It fails when the number of possible outcomes of the experiment is infinite.  It is based on the cases which are “equally likely” and as such cannot be applied to experiments where the outcomes are not equally likely. 14  It may not be possible practically to enumerate all the possible outcomes of certain experiments and in such cases the method fails. Example it is inadequate for answering questions such as: What is the probability that a man aged 45 will die within the next year? Here there are only 2 possible outcomes, the individual will die in the ensuing year or he will live. The chances that he will die is of course much smaller than he will live. How much smaller? 15 Combinations This counting rule for combinations allows us to select r (say) number of outcomes from a collection of n distinct outcomes without caring in what order they are arranged. This rule is denoted by 16  Example 5.1: Of 10 electric bulbs, 3 are defective but it is not known which are defective. In how many ways can 3 bulbs be selected? How many of these selections will include at least 1 defective bulb? Example 5.2: A bag contains 6 red and 8 green balls. (a) If one ball is drawn at random, then what is the probability of the ball being green? (b) If two balls are drawn at random, then what is the probability that one is red and the other green? 17  2 A card is drawn from a well-shuffled deck of 52 cards. Find the probability of drawing a card which is neither a heart nor a king. 5.3 In a single throw of two dice, find the probability of getting (a) a total of 11, (b) a total of 8 or 11, and (c) same number on both the dice. 5.4 Five men in a company of 20 are graduates. If 3 men are picked out of the 20 at random, what is the probability that they are all graduates? What is the probability of at least one graduate? A bag contains 25 balls numbered 1 through 25. Suppose an odd number is considered a “success”. Two balls are drawn from the bag with replacement. Find the probability of getting (a) two successes (b) exactly one success (c) at least one success (d) no successes 18 Permutations This rule of counting involves ordering or permutations. This rule helps us to compute the number of ways in which n distinct objects can be arranged, taking r of them at a time. The total number of permutations of n objects taken r at a time is given by 19  Example 5.3: Tickets are numbered from 1 to 100. They are well shuffled and a ticket is drawn at random. What is the probability that the drawn ticket has (a) an even number? (b) the number 5 or a multiple of 5? (c) a number which is greater than 75? (d) a number which is a square? A bag contains 5 white and 8 red balls. Two drawings of 3 balls are made such that (a) the balls are replaced before the second trial and (b) the balls are not replaced before the second trial. Find the probability that the first drawing will give 3 white and the second will give 3 red balls in each case. 20 Theorems of Probability There are 2 important theorems of probability which are as follows:  The Addition Theorem and  The Multiplication Theorem 21 Addition theorem when events are Mutually Exclusive Definition: - It states that if 2 events A and B are mutually exclusive then the probability of the occurrence of either A or B is the sum of the individual probability of A and B. Symbolically P(A or B) or P(A U B) = P(A) + P(B) The theorem can be extended to three or more mutually exclusive events. Thus, P(A or B or C) = P(A) + P(B) + P(C) 22 23 Addition theorem when events are not Mutually Exclusive (Overlapping or Intersection Events) Definition: - It states that if 2 events A and B are not mutually exclusive then the probability of the occurrence of either A or B is the sum of the individual probability of A and B minus the probability of occurrence of both A and B. Symbolically P(A or B) or P(A U B) = P(A) + P(B) – P(A ∩ B) 24 Q1 Example What is the probability that a randomly chosen card from a deck of cards will be either a king or a heart. Q2 Example Of 1000 assembled components, 10 have a working defect and 20 have a structural defect. There is a good reason to assume that no component has both defects. What is the probability that randomly chosen component will have either type of defect? Q3 From a computer tally based on employer records, the personnel manager of a large manufacturing firm finds that 15 per cent of the firm’s employees are supervisors and 25 per cent of the firm’s employees are college graduates. He also discovers that 5 per cent are both supervisors and college graduates. Suppose an employee is selected at random from the firm’s personnel records, what is the probability of (a) selecting a person who is both a college graduate and a supervisor? (b) selecting a person who is neither a supervisor nor a college graduate? 25  An MBA applies for a job in two firms X and Y. The probability of his being selected in firm X is 0.7 and being rejected at Y is 0.5. The probability of at least one of his applications being rejected is 0.6. What is the probability that he will be selected by one of the firms? The probability that a contractor will get a plumbing contract is 2/3 and the probability that he will not get an electrical contract is 5/9. If the probability of getting at least one contract is 4/5, what is the probability that he will get both? 26 Multiplication theorem Definition: States that if 2 events A and B are independent, then the probability of the occurrence of both of them (A & B) is the product of the individual probability of A and B. Symbolically, Probability of happening of both the events: P(A and B) or P(A ∩ B) = P(A) x P(B) Theorem can be extended to 3 or more independent events. Thus, P(A, B and C) or P(A ∩ B ∩ C) = P(A) x P(B) x P(C) 27  Rules of Multiplication Statistically Independent Events When the occurrence of an event does not affect and is not affected by the probability of occurrence of any other event, the event is said to be a statistically independent event. There are three types of probabilities under statistical independence: marginal, joint, and conditional. Marginal Probability: A marginal or unconditional probability is the simple probability of the occurrence of an event. For example, in a fair coin toss, the outcome of each toss is an event that is statistically independent of the outcomes of every other toss of the coin. Joint Probability: The probability of two or more independent events occurring together or in succession is called the joint probability. The joint probability of two or more independent events is equal to the product of their marginal probabilities. In particular, if A and B are independent events, the probability that both A and B will occur is given by 28  Suppose we toss a coin twice. The probability that in both the cases the coin will turn up head is given by 29 30  Q1 The odds against student X solving a Business Statistics problem are 8 to 6 and odds in favour of student Y solving the problem are 14 to 16. (a) What is the chance that the problem will be solved if they both try independently of each other? (b) What is the probability that none of them is able to solve the problem? Q2 The probability that a new marketing approach will be successful is 0.6. The probability that the expenditure for developing the approach can be kept within the original budget is 0.50. The probability that both of these objectives will be achieved is 0.30. What is the probability that at least one of these objectives will be achieved. For the two events described above, determine whether the events are independent or dependent. Q3 A piece of equipment will function only when the three components A, B, and C are working. The probability of A failing during one year is 0.15, that of B failing is 0.05, and that of C failing is 0.10. What is the probability that the equipment will fail before the end of the year? 31 32 33  A company has two plants to manufacture scooters. Plant I manufactures 80 per cent of the scooters and Plant II manufactures 20 per cent. In plant I, only 85 out of 100 scooters are considered to be of standard quality. In plant II, only 65 out of 100 scooters are considered to be of standard quality. What is the probability that a scooter selected at random came from plant I, if it is known that it is of standard quality? The probability that a trainee will remain with a company is 0.6. The probability that an employee earns more than Rs. 10,000 per month is 0.5. The probability that an employee who is a trainee remained with the company or who earns more than Rs. 10,000 per month is 0.7. What is the probability that an employee earns more than Rs. 10,000 per month given that he is a trainee who stayed with the company? 34  Two computers A and B are to be marketed. A salesman who is assigned the job of finding customers for them has 60 per cent and 40 per cent chances of succeeding for computers A and B, respectively. The two computers can be sold independently. Given that he was able to sell at least one computer, what is the probability that computer A has been sold? A committee of 4 persons is to be appointed from 3 officers of the production department, 4 officers of the purchase department, two officers of the sales department and 1 chartered accountant. Find the probability of forming the committee in the following manner : (i) There must be one from each category (ii) It should have at least one from the purchase department (iii) The chartered accountant must be in the committee. 35 How to calculate probability in case of Dependent Events Case Formula 1. Probability of occurrence of at least A or B 1. When events are mutually P(A U B) = P(A) + P(B) 2. When events are not mutually exclusive P(A U B) = P(A) + P(B) – P(A ∩ B) 2. Probability of occurrence of both A & B P(A ∩ B) = P(A) + P(B) – P(A U B) 3. Probability of occurrence of A & not B P(A ∩ B) = P(A) - P(A ∩ B) 4. Probability of occurrence of B & not A P(A ∩ B) = P(B) - P(A ∩ B) 5. Probability of non-occurrence of both A & B P(A ∩ B) = 1 - P(A U B) 6. Probability of non-occurrence of atleast A or B P(A U B) = 1 - P(A ∩ B) 36 How to calculate probability in case of Independent Events Case Formula 1. Probability of occurrence of both A & B P(A ∩ B) = P(A) x P(B) 2. Probability of non-occurrence of both A P(A ∩ B) = P(A) x P(B) &B P(A ∩ B) = P(A) x P(B) 3. Probability of occurrence of A & not B P(A ∩ B) = P(A) x P(B) 4. Probability of occurrence of B & not A P(A U B) = 1 - P(A ∩ B) = 1 – [P(A) x P(B)] 5. Probability of occurrence of atleast one event P(A U B) = 1 - P(A ∩ B) = 1 – [P(A) x P(B)] 6. Probability of non-occurrence of atleast one event P(A ∩ B) + P(A ∩ B) = [P(A) x P(B)] + 7. Probability of occurrence of only one event [P(A) x P(B)] 37  Revising Prior Estimates of Probabilities: Bayes’ Theorem A very important & useful application of conditional probability is the computation of unknown probabilities, based on past data or information. When an event occurs through one of the various mutually disjoint events, then the conditional probability that this event has occurred due to a particular reason or event is termed as Inverse Probability or Posterior Probability. Has wide ranging applications in Business & its Management. 38  Since it is a concept of revision of probability based on some additional information, it shows the improvement towards certainty level of the event. Example 1: - If a manager of a boutique finds that most of the purple & white jackets that she thought would sell so well are hanging on the rack, she must revise her prior probabilities & order a different color combination or have a sale. Certain probabilities were altered after the people got additional information. New probabilities are known as revised, or Posterior probabilities. 39 Bayes Theorem If an event A can occur only in conjunction with n mutually exclusive & exhaustive events B1, B2, …, Bn, & if A actually happens, then the probability that it was preceded by an event Bi (for a conditional probabilities of A given B1, A given B2 … A given Bn are known) & if marginal probabilities P(Bi) are also known, then the posterior probability of event Bi given that event A has occurred is given by: 40  Example 5.24: Suppose an item is manufactured by three machines X, Y, and Z. All the three machines have equal capacity and are operated at the same rate. It is known that the percentages of defective items produced by X, Y, and Z are 2, 7, and 12 per cent, respectively. All the items produced by X, Y, and Z are put into one bin. From this bin, one item is drawn at random and is found to be defective. What is the probability that this item was produced on Y? Assume that a factory has two machines. Past records show that machine 1 produces 30 per cent of the items of output and machine 2 produces 70 per cent of the items. Further, 5 per cent of the items produced by machine 1 were defective and only 1 per cent produced by machine 2 were defective. If a defective item is drawn at random, what is the probability that the defective item was produced by machine 1 or machine 2? In a bolt factory, machines A, B, and C manufacture 25 per cent, 35 per cent and 40 per cent of the total output, respectively. Of the total of their output, 5, 4, and 2 per cent are defective bolts. A bolt is drawn at random and is found to be defective. What is the probability that it was manufactured by machines A, B, or C? 41  A petrol pump proprietor sells on an average Rs. 80,000 worth of petrol on rainy days and an average of Rs. 95,000 on clear days. Statistics from the Metrological Department show that the probability is 0.76 for clear weather and 0.24 for rainy weather on coming Monday. Find the expected value of petrol sale on coming Monday. (a) Given that four airlines provide service between Delhi and Mumbai. In how many distinct ways can a person select airlines for a trip from Delhi to Mumbai and back, if (i) he must travel both the ways by the same airline and (ii) he travel from Delhi to Mumbai by one airline and returns by another? (b) 6 coins are tossed simultaneously. What is the probability that they will fall with 4 heads and 2 tails up? (c) If A is the event of getting a prize on the first punch and B the event of getting the prize on the second punch, calculate the probability of getting two prizes taking two punches on a punch board which contains 20 blanks and five prizes? (d) In how many ways can the word ‘PROBABILITY’ be arranged? (e) What is the probability of getting a number greater than two with an ordinary dice? 42  A manufacturing firm produces steel pipes in three plants with daily production volumes of 500, 1000, and 2000 units respectively. According to past experience, it is known that the fractions of defective output produced by the three plants are respectively.005,.008, and.010. If a pipe is selected from a day’s total production and found to be defective, find out (a) from which plant the pipe comes? (b) What is the probability that it came from the first plant? Two dice are thrown at a time. Let E1 be the event of getting 6 on the first dice and E2 that of getting 1 on the second dice. Are the events E1 and E2 independent? A card is drawn from a well-shuffled deck of 52 cards. Find the probability of drawing a card which is neither a heart nor a king. 43  A can solve 90 per cent of the problems given in a book and B can solve 70 per cent. What is the probability that at least one of them will solve a problem selected at random? A bag contains 6 white, 4 red, and 10 black balls. Two balls are drawn at random. Find the probability that they will both be black. (a) A problem of statistics is given to two students A and B. The odds in favour of A solving the problem are 6 to 9 and against B solving the problem 12 to 10. If A and B attempt, find the probability of the problem being solved A committee of 4 people is to be appointed from 3 officers of the production department, 4 officers of the purchase department, 2 officers of the sales department, and 1 chartered accountant. Find the probability of forming the committee in the following manner : (a) There must be one from each category. (b) It should have at least one from the purchase department. (c) The chartered accountant must be on the committee. 44 Remarks: -  The probabilities P(B1), P(B2), … , P(Bn) are termed as the ‘a priori probabilities’ because they exist before we gain any information from the experiment itself.  The probabilities P(A | Bi), i=1,2,…,n are called ‘Likelihoods’ because they indicate how likely the event A under consideration is to occur, given each & every a priori probability.  The probabilities P(Bi | A), i=1, 2, …,n are called ‘Posterior probabilities’ because they are determined after the results of the experiment are known. 45 Glossary of terms Classical Probability: It is based on the idea that certain occurrences are equally likely.  Example: - Numbers 1, 2, 3, 4, 5, & 6 on a fair die are each equally likely to occur. Conditional Probability: The probability that an event occurs given the outcome of some other event. Independent Events: Events are independent if the occurrence of one event does not affect the occurrence of another event. Joint Probability: Is the likelihood that 2 or more events will happen at the same time. Multiplication Formula: If there are m ways of doing one thing and n ways of doing another thing, there are m x n ways of doing both. 46  Mutually exclusive events: A property of a set of categories such that an individual, object, or measurement is included in only one category. Objective Probability: It is based on symmetry of games of chance or similar situations. Outcome: Observation or measurement of an experiment. Posterior Probability: A revised probability based on additional information. Prior Probability: The initial probability based on the present level of information. Probability: A value between 0 and 1, inclusive, describing the relative possibility (chance or likelihood) an event will occur. Subjective Probability: Synonym for personal probability. Involves personal judgment, information, intuition, & other subjective evaluation criteria.  Example: - A physician assessing the probability of a patient’s recovery is making a personal judgment based on what they know and feel about the situation. 47