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- Introduction to probability MAHESH MT ASSISTANT PROFESSOR OF COMMERCE KNM GOVT. ARTS AND SCIENCE COLLEGE
- Introduction to Probability Certain Possible and Impossible Events Sample space The compliment of an event Mutually exclusive events Addition and multiplication rules for probability. Independent and dependent events. Combinatorial probability Conditional probability Bayes theorem/ Law of total probability Central limit theorem CONTENTS
- “It is likely to rain today.” “I have a fair chance of getting admission.” “There is an even chance that in tossing a coin, the head may come up.” In each case, we are not certain of the outcome, but we wish to assess the chances of our predictions coming true. If we assign numerical value, the statement would become more precise. The theory of probability provides a numerical measure of the element of uncertainty. Probability refers to the chance of happening or not happening an event. Introduction to Probability
- Experiment – It denotes to describe an act which can be repeated under some given conditions. Random experiment: An experiment has two or more outcomes which may vary in an unpredictable manner from trial to trial when conducted under uniform conditions. It is called ‘random experiment.’ Trial : Performing a random experiment. (Tossing a coin) Outcome: This is the result of a random experiment.( T) Event: An outcome or a combination of outcomes of a random experiment .( HH) Terms used in probability
- The theory of probability has its origin in the ‘games of chance ’’ related to gambling such as tossing a coin a, drawing cards from a pack of cards etc. Jerame Cardon (1501-76), an Italian mathematician, was the first person who wrote a book named ‘Book on games of chance’. Starting with the games of chance, probability has become one of the important statistical tool today. A knowledge of probability is required to interpret statistical results as many statistical procedures involve conclusions based on samples. origin
- sample point/ sample element/elementary outcome: Every indecomposable outcome of a random experiment is called a sample point of that random experiment. Eg: tossing a coin and getting Head is a sample point. Sample space: It is the set which contains all the sample points of that random experiment. It is totality of all the elementary outcome of that experiment. A sample space may be discrete or continuous in nature. Eg: tossing a coin twice, the sample space is HH, HT,TH and TT denotes as S={HH, HT,TH, TT} Terms used in probability
- Certain event : An event whose occurrence is inevitable is called certain event. Eg: Getting a blue ball from a bag containing all blue balls. Impossible event: - in the above example, getting a black ball from the set of blue balls is an impossible event. An empty set can be denotes as ø. Ø= null set. Uncertain event:- it is the event which cannot predict its happening. The happening of that event is neither sure nor impossible. Eg: Getting a black ball from a bag containing white and black balls. It is uncertain. Terms used in probability
- Equally likely events:- Two or more events are said to be equally likely or equally probable if each one of them has equal chance of occurring. Eg: while tossing a coin, getting a Head or tail has equal chance and hence it is equally likely. Mutually exclusive events: Two or more events are said to be mutually exclusive if the happening of any one of them excludes the happening of all others in the same experiment. Eg: when a coin is tossed, the occurrence of Head completely excludes the occurrence of tail. Terms used in probability
- Exhaustive events: A group of events is said to be exhaustive event, when it includes all possible outcomes of the random experiment under consideration. Eg: while tossing a single coin, the outcome will be head /tail. Head and Tail together will form an exhaustive event because one of them will occur when a coin is tossed. Mutually exclusive and exhaustive events/ complementary events: A set of events said so, if one of them must and only one can occur. Eg: while tossing two coins, getting no head or getting one head or getting two heads are mutually exclusive and exhaustive. Terms used in probability
- Compound events : refers to the joint occurrence of two or more events. Thus compound events imply the simultaneous occurrence of two or more simple events. Independent events: Two or more events are said to be independent if the occurrence of one of them in no way effects the occurrence of other event. Eg: tossing a coin twice. The result of second tossing is not effected by the result of first tossing. Dependent events: The happening of one event in any one trial affects the probability of other events in other trials. Eg: Drawing a ‘King’ first from a packet of cards, then draws another king without replacing the first king in the packet of cards. The second event is dependent event. Terms used in probability
- Meaning : The number of occasions that a particular event is likely to occur in a large population of events. A particular event can be expressed positively where the event is likely to happen and negatively when the event is not likely to happen. Ranges from 0 to 1. Definition :if the probability can be calculated even before the actual happening of the event, ie even before conducting the experiment is mathematical probability. if the probability can be determined only after the actual happening of the event, ie after conducting the experiment is statistical probability. Meaning and Definition of probability
- 1. objective Probability a) Classical /priori / mathematical probability ( the outcomes of a random experiment are equally likely) b) Empirical / Posteriory / Statistical Probability ( used when it is not possible to make a simple enumeration of a case which can be considered equally likely) 2. Subjective / Personalistic Probability ( Probability as a measure of personal confidence in a particular proposition) 3. Modern / Axiomatic Approach to Probability (combines both subjective and objective concepts ) CONCEPTS USED IN PROBABILITY
- The probability is the ratio of the number of favourable cases to the total number of equally likely cases – Laplace (French mathematician) This is also referred as ‘classical probability’ and ‘priori probability’. Probability of occurrence P(A)=success= 𝑚 𝑛 = 𝑛𝑜.𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑡𝑜𝑡𝑎𝑙 𝑛𝑜.𝑜𝑓 𝑐𝑎𝑠𝑒𝑠 Probability of non occurrence Q =failure =P(not A) = 𝑛−𝑎 𝑛 This formula may not be applied under the following situations 1. if it is not possible to enumerate all possible outcomes. 2. If the sample points are not mutually independent. 3. If the total number of items are infinite. 4. If each and every outcome is not equally likely. Note: Priori means “from cause to effect’’. 1. Mathematical probability
- Q. From a bag containing 10 black and 20 red balls , a ball is drawn at random. 1. What is the probability that it is a black ball ? 2. What is the probability that it is not a black ball ? Solution 1. 1. total number of cases =10+20=30 2.number of black balls =10 (favourable cases) Probability of getting a black ball q= m/n= 10/30=1/3 Solution 2 Probability of not getting a black ball= n-m/n =30-10/30=20/30=2/3 Then p + q = 1/3 + 2/3 =1 Example
- If the probability of an event can be determined only after the actual happening of an event. This approach was initiated by the mathematician VON MISES used when it is not possible to make a simple enumeration of a case which can be considered equally likely. Probability of an event can here be defined as the relative frequency with which it occurs on an indefinitely large number of trials . if an event occurs ‘m’ times out of ‘n’ times ,its relative frequency is m/n in the limiting case , when ‘n’ becomes sufficiently large it corresponds to a number which is called the probability of an event. Here P(A)= lim 𝑛→∞ (𝑚/𝑛) When this experiment is carried out a large number of times, the expected outcome approached to 0.5 2. Statistical probability/ posteriori
- Subjective theorists regard Probability as a measure of personal confidence in a particular proposition. Eg: A belief that a particular students of a college will secure rank in the university examination . Here the subjectivist can assign a weight between zero and one to an even according to his belief. It may vary from person to person , hence it is personalistic probability EG: I BELIEVE THAT YOU ALL WILL PASS THIS EXAM Subjective approach
- A,B,C are used to denote sets , let A be the first set and B be another set. Where A =(3,4,6,7) and B = (1,2,3,4,5) Thus A and B or A U B (A Union B)=(1,2,3,4,5,6,7) The event which consists of all points that are in A or B or in both is called the union of A and B. It is written as A U B INTERSECTION : The event which includes all points which are common in both sets is called 𝐴 ∩ 𝐵 ( A Intersection B) Here 3 and 4 are in both sets SET THEORY
- • This approach is developed by Russian Mathematician A.N Kolmogrove in the year 1933. • It combines both subjective and objective probability • When this approach is followed , no precise definition of probability is given ,rather we give certain axioms or postulates on which probability calculations are based • 1. The Probability of an event ranges from zero to one. If the event cannot take place, the probability will be zero If the event take place, the probability shall be one symbolically 0≤P(A)≤1 • 2. The sum of all probabilities of all sample events constituting that sample space is equals to 1 ie, P(S)=1 • 3. The probability of compound event is the sum of the probabilities of sample events. If A and B are mutually exclusive, then Probability of either A or B is denoted by P(A U B) = P(A)+P(B) Axiomatic /modern Approach
- The event which includes all parts which are not in ‘A’ is called a compliment of A and written as Ā The shaded area on the diagram indicates Ā Universe of Discourse – the set containing all elements under discussion for a particular problem. In mathematics, this is called the universal set and is denoted by U. Venn diagrams can be used to represent sets and their relationships to each other . Complementation
- SET THEORY Interpretation of statistical statements in terms of set theory: S = Sample Space = 𝐴 ∪ Ā A => A does not occur 𝐴 ∩ Ā = 𝐴 𝑎𝑛𝑑 𝐵 𝑎𝑟𝑒 𝑚𝑢𝑡𝑢𝑎𝑙𝑙𝑦 𝑒𝑥𝑐𝑙𝑢𝑠𝑖𝑣𝑒 A U B = Event A occurs or B occurs or both A and B occurs. 𝐴 ∩ 𝐵 = 𝐵𝑜𝑡ℎ 𝑒𝑣𝑒𝑛𝑡𝑠 𝐴 𝑎𝑛𝑑 𝐵 𝑜𝑐𝑐𝑢𝑟𝑠 Ā ∩ 𝐵 = Neither A nor B occurs A ∩ 𝐵 = Event A occurs and B does not occur Ā ∩ 𝐵 = 𝐸𝑣𝑒𝑛𝑡 𝐴 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑜𝑐𝑐𝑢𝑟 𝑎𝑛𝑑 𝐵 𝑜𝑐𝑐𝑢𝑟𝑠.
- 1. The Addition theorem •Mutually exclusive events •Not mutually exclusive events 2. The Multiplication theorem •Independent events •Dependent events Theories of Probability
- The Addition Theorem It states that if two events A and B are mutually exclusive , the probability of the occurrence of ‘A’ and ‘B’ is the sum of the individual probability of A and B. P (A 0r B)= P(A)+P(B) Technically P(A ∪ 𝐵) = 𝑃 𝐴 + 𝑃 𝐵 Example : The probability that student X will s0lve a particular problem is 1/8 and student Y will solve the same problem is ¼. What is the probability that either A or B will solve the problem? Solution: let’s A =event of solving the problem by X =P(A)=1/8 B= event of solving the problem by Y =P(B)= ¼ Therefore P (𝐴 ∪ 𝐵) =1/8 + ¼ = 4+8/32 =12/32=0.375
- Addition theorem for any two events (not mutually exclusive) If events are not mutually exclusive , then probability for A or B to happen is the sum of their probabilities minus probability of both to happen P (A 0r B)= P(A)+P(B) –P(A&B) P(A ∪ 𝐵) = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵) Example :find the probability of drawing an Ace or Club from a normal deck of cards. P(A)= Event of Ace occuring =4/52 , P(B)= Event of club =13/52 The event of club Ace occuring = P (𝐴 ∩ 𝐵) =1/52 Then 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵) Ie. 4/52 + 13/52 – 1/52 = 16/52 The Addition Theorem
- Multiplication Theorem It states that, if two events A and B are dependent , the probability of the second event occuring will be effected by the outcome of the first draw that has already occurred. Symbolically P(A ∩ 𝐵) = 𝑃(𝐴). 𝑃(𝐵/𝐴) Example : a bag contains 5 white balls and 8 black balls . One ball is drawn at random from the bag and then it is not replaced. Again another one is drawn. Find the probability that both the balls drawn are white? Solution: P(A)= Event occuring white ball in first draw = 5/13 P(B/A )= Event occuring white ball in Second draw = 4/12 P of occuring white ball in both draws = P(A∩ 𝐵) = 𝑃(𝐴). 𝑃(𝐵/𝐴) Ie , 5/13 x 4/12 =20/156
- Multiplication Theorem It states that if two events A and B are independent , the probability that they will both occur is equal to the product of their individual probabilities. P (A&B)=P(A) X P(B) Technically P(A ∩ 𝐵) = 𝑃(𝐴). 𝑃(𝐵) P(A ∩ 𝐵 ∩ 𝐶) = 𝑃(𝐴). 𝑃(𝐵). 𝑃(𝐶) Example : a bag contains 5 white balls and 8 black balls . One ball is drawn at random from the bag and then replaced. Again another one is drawn. Find the probability that both the balls drawn are white? Solution: P(A)= Event occuring white ball in first draw = 5/13 P(B )= Event occuring white ball in Second draw = 5/13 P of occuring white ball in both draws = P(A∩ 𝐵) = 𝑃(𝐴). 𝑃(𝐵) Ie , 5/13 x 5/13 =25/169
- If two events A and B are dependent. Event B can occur only when event A is known to have occurred or vice versa. The probability attached to such an event is called conditional probability and is denoted as P(B/A) If both events are dependent , Probability of B given A is P(B/A) = 𝑃(𝐴 ∩ 𝐵)/𝑃(𝐴) If both events are dependent , Probability of A given B is P(A/B) = 𝑃(𝐴 ∩ 𝐵)/𝑃(𝐵) CONDITIONAL PROBABILITY
- PRACTICE QUESTIONS Q2. When two dice are thrown, find the probability of getting doublets ( same number on both dice) S= ( 6x6)=36
- PRACTICE QUESTIONS Q1. Three coins are tossed simultaneously , find the probability that 1) no head 2) one head 3) two heads 4) At least two heads S= ( HHH, HHT, HTT, TTT,THT,THH,HTH,TTH)
- Contribution of Sir Thomas Baye’s ,an English Philosopher His contribution consists of a primarily a unique method for calculating conditional probabilities. Bayes theorem is based on the preposition that probabilities should be revised when the information is available. Ie the occurrence of one event to predict the occurrence of another. The need for revising probabilities arise from a need to make better use of available information. This will help to reduce the element of risk involved in decion making the probabilities before revision is called priori probabilities and those after revision is called posterior probabilities BAYES THEOREM
- 1. Under this principle , by assuming certain priori probabilities, the posteriori probabilities are obtained. So the Baye’s theorem are also called posteriori probabilities. Rule : According to Baye’s theorem, the probability of event (A) for a particular result of an investigation (B) may be found from P(A/B) = P(A) P(B/A) P(A) P(B/A)+P(Not A) P(B/Not A) Similarly P(B/A) = P(B) P(A/B) P(B) P(A/B)+P(Not B) P(A/Not B) BAYES THEOREM
- EX: A is known to tell the truth five times in cases out of six. He the states that a white ball was drawn from a bag 0f 9 black and one white ball. What is the probability that the white ball is really drawn? Solution : The probability that the white ball is drawn =1/10 The Probability that white ball is drawn and A told the truth = 1/10 x 5/6 The probability that a black ball was drawn and A told a lie =9/10 x 1/6 The probability that a white ball was drawn P(A) ,given B P(A/B) = P(A) P(B/A) = 1/10 X 5/6 = 5 P(A) P(B/A)+P(Not A) P(B/Not A) (1/10 X5/6) + ( 9/10X1/6) 14 BAYES THEOREM
- CLT in probability theory states that , given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well defined expected value and well defined variance, will be approximately normally distributed, regardless of the underlying distribution. CLT states that the distribution of sample approximates a normal distribution (also known as a bell curve) as the sample size becomes larger, assuming all samples are identical in size regardless of the population distribution shape Examples : laboratory measurement errors are usually modeled by normal random variables. in finance, percentage of changes in prices of some assets are sometimes modeled by normal random variables Central limit theorem
- Fundamental principle If one thing can be done in ‘m’ different ways and a second thing can be done in ‘n’ different ways. Thus the two things can be done ‘m’ ‘n’ different ways Factorial symbol nỊ = n factorial denotes the product nỊ = n(n-1)Ị (n+1)Ị = (n+1)nỊ Ex: 6Ị = 6x5x4x3x2x1= 720 Permutation and combination
- It refers to different arrangements of objects in a set when all elements are different and distinguishable. The arrangements are done without repetition of an individual object It refers to separate arrangements of different objects contained in a set of elements. Example : if a factory owner receives 3 different machines, A B and C. how many ways it can be ? ABC, ACB, BCA,BAC,CBA,CAB nỊ= 3Ị= 3x2x1=6 times Permutation
- When we have ‘n’ different objects and ‘r’ space is to be filled, then by the multiplication rule, the space can be filled in as follows n(n-1)(n-2)(n-3)……. Let npr = n(n-1)(n-2)…..(n-r+1) By using factorial symbols Npr = n(n-1)(n-2)…(n-r+1)(n-r)Ị = nỊ (n-r)Ị (n-r)Ị Ex: indicate how many ways 5 letter words can be formed from the letters in the word ‘EQUATIONS’ SOLUTION; here n= 9, r=5 npr = nỊ = 9p5= 9Ị = 9x8x7x6x5x4x3x2x1 = 9X8X7X6X5 (n-r)Ị (9-5)Ị 4x3X2x1 = 15,120 Permutation
- A combination is a selection if objects. Combination is a grouping or selection of all or part of a number of things without reference to the arrangement of the thing selected. The number of combination of objects are different from their permutation The combination of the letter A B and C are ABC The total number of possible combinations of a set of objects taken at a time is 1. only one combination is possible for a set. The number of combinations of different objects taken ‘r’ at a time is denoted as ncr = nỊ rỊ (n-r)Ị In combination ,we are not concerned with the arrangements of objects . In permutations , a combination can give rise to several permutations Ex: combination ABC , Permutation: ABC,ACB,BAC,BCA,CAB,CBA combination
- How many different sets of 5 students can be chosen out of 20 qualified students to represent a college in a quiz competion? n = 20 (no. of qualified students ) r = 5 ( no. of students to be selected at a time) Ncr = nỊ rỊ (n-r)Ị 20c5 = 20Ị = 15504 5Ị (20-5)Ị combination
- In how many ways can 4 red balls be drawn from a bag containing 10 red balls n = 10 (no. of red balls ) r = 4 ( no. of red balls to be selected at a time) Ncr = nỊ rỊ (n-r)Ị 10c4 = 10Ị =210 4Ị (10-4)Ị combination

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