Probability Dr. Manjula Gunarathna 1

Transcription

1 Probability Dr. Manjula Gunarathna Probability Dr. Manjula Gunarathna 1

2 Introduction Probability theory was originated from gambling theory Probability Dr. Manjula Gunarathna 2

3 History of Probability Galileo ( ) an Italian mathematician- first man to attempt quantitative measure of probability B. Pascal ( ) and Pierre de Fermat ( ) two French mathematicians-systematic and scientific foundation of the mathematical theory of probability James Bernoulli ( ) Swiss mathematiciantreatise on probability published De Moivre ( ) Dotcrines of Charles published in 1718 Probability Dr. Manjula Gunarathna 3

4 Thomas Bayes ( ) - Inverse Probability Pierre-Simon de Laplace ( )- Theory of Analytical Probability R.A. Fisher and Von Mises - empirical approach to probability Chebychev ( ) and A. kolmogorov. - Russian mathematicians - modern theory of probability Probability Dr. Manjula Gunarathna 4

5 The word probability or chance is commonly used in our daily conversation and we have a vague idea about its meaning Consider the following day-to-day life statements 1. Everyday Sun rises in the east 2. It is possible to live without water 3. Probably Nimal gets that job Impossibility (2) P=0 Positive fraction (3) 0 p 1 Certainty (1) P=1 Probability Dr. Manjula Gunarathna 5

6 The utility and importance of probability in Economics Predictions for future It is very much used in economic decision making It is extensively used in economic situations characterized by uncertainty (viz, investment problem, inventory problem, problem of introducing new product and so on) It is the base of the fundamental laws of economics i.e. decision theory Probability Dr. Manjula Gunarathna 6

7 Definition of probability The probability when defined in the simplest way is chance or occurrence of a certain event when expressed quantitatively. The probability is defined in four different ways though its approaches 1. Subjective (personalistic) approach 2. Classical (a priori) approach 3. Statistical (empirical) approach 4. Axiomatic (modern) approach Probability Dr. Manjula Gunarathna 7

8 Subjective (personalistic) approach This approach is used to determine the probability of events which have either not occurred at all in the past or which occur only once or where experiment cannot be performed repeatedly under identical conditions. J.M. Keynes and L.J. Savage have identified the subjective probability as a measure of one s confidence in the occurrence of a particular event. Probability Dr. Manjula Gunarathna 8

9 Classical (a priori) approach If an experiment has n mutually exclusive, equally likely and exhaustive cases, out of which m are favorable to the happening of event A, then the probability of the happening of A is denoted by P (A) and is defined as; P(A) = m/n = No. of favorable to A/Total number of cases Probability Dr. Manjula Gunarathna 9

10 Statistical (empirical) approach Von Mises has give the following statistical definition. if the experiment be repeated a large number of times under essentially identical conditions, the limiting values of the ratio of the number of times the event E happens to the total number of trials of the experiment as the number of trials increases indefinitely is called the probability of happening of the E Probability Dr. Manjula Gunarathna 10

11 Symbolically n P(E) = Lim n N Let P(E) denote the probability of the occurrence of E. Let n be the number of times in which an event E occurs in a series of N trials. Probability Dr. Manjula Gunarathna 11

12 Axiomatic (modern) approach The Russian mathematician A.N. Kolmogorv introduced this new modern approach through the theory of sets in The modern definition of probability includes both the classical and the statistical definitions as particular cases overcomes the deficiencies of each of them. It is based on certain axioms. The advantage of the axiomatic theory is that it narrates all situations irrespective of whether the outcomes of an experiment are equally likely or not. Probability Dr. Manjula Gunarathna 12

13 (i) for all i 0 p(s i ) 1 (ii) p (s i ) = 1 Probability Dr. Manjula Gunarathna 13

14 Some Important Terms and Concepts Experiment The term experiment refers to processes which result in different possible out come or observation. Ex; tossing a coin, or throwing a dice Probability Dr. Manjula Gunarathna 14

15 Sample Space (S) A set of all possible outcomes from an experiment is called a Sample Space. Let us toss a coin, the result is either head or tail. Let 1 denote head and 0 denote tail. S={0"1} throwing a dice S={1"2"3"4"5"6} Probability Dr. Manjula Gunarathna 15

16 Mark the point 0, 1 on a Straight line. These Points are called Sample Points or Event Points. For a given experiment there are different possible outcomes and hence different sample points. The collection of all such sample points is a Sample Space. Probability Dr. Manjula Gunarathna 16

17 Discrete Sample Space A sample Space whose elements are finite or infinite but countable is called a discrete Sample Space. For example, if we toss a coin as many times as we require for turning up one head, then the sequence of points S 1 =(1), S 2 = (0,1), S 3 = (1,0,0), S 4 = (0,0,0,1) etc., is a discrete Sample Space. Probability Dr. Manjula Gunarathna 17

18 Continuous Sample Space A sample space whose elements are infinite and uncountable or assume all the values on a real line R or on an interval of R is called a Continuous Sample Space. In this case the sample points build up a continuum, and the sample space is said to be continuous. Probability Dr. Manjula Gunarathna 18

19 coin Graph T H legh dice Probability Dr. Manjula Gunarathna 19

20 Figure H T Probability Dr. Manjula Gunarathna 20

21 Event A sub-collection of a number of sample points under a definite rule or law called an event. For example, let us take a dice. Let its faces 1,2,3,4,5,6 be represented by E 1,E 2,E 3,E 4,E 5,E 6 respectively. Then all the E i s are sample points. Let E be the event of getting an even number on the dice. Obviously, E= {E 2, E 4, E 6 }, which is a subset of the set {E 1, E 2, E 3, E 4, E 5, E 6 } Probability Dr. Manjula Gunarathna 21

22 Null Event An event having no sample point is called a null event Probability Dr. Manjula Gunarathna 22

23 Simple Event An event consisting of only one sample point of a sample space is called a simple event. For example, let a dice be rolled once and A be the event that face number 5 is turned up, then A is a Simple event. Probability Dr. Manjula Gunarathna 23

24 Compound Events When an event is decomposable into a number of simple events, then it is called a compound event. For example, the sum of the two numbers shown by the upper faces of the two dice is seven in the simultaneous throw of the two unbiased dice, is a compound event as it can be decomposable. Probability Dr. Manjula Gunarathna 24

25 Equally Probable Events. If in an experience all possible outcomes have equal chances of occurrence, then such events are said to be equally probable events. For example, in throwing a coin, the events head and tail have equal chances of occurrence, therefore, they are equally probable events. Probability Dr. Manjula Gunarathna 25

26 Favourable Cases The cases which ensure the occurrence of an event are said to be favourable to the event. Probability Dr. Manjula Gunarathna 26

27 Independent and Dependent Events Two or more events are said to be independent if the happening of any one does not depend on the happening of the other. Events which are not independent are called dependent events. Probability Dr. Manjula Gunarathna 27

28 Probability Rules Addition Rule Multiplication Bayes Rule Events are mutually exclusive Events are not mutually exclusive Events are independent Events are dependent Probability Dr. Manjula Gunarathna 28

29 Mutually Exclusive Events If in an experiment the occurrence of an event prevents or rules out the happening of all other events in the same experiment, then these events are said to be Mutually Exclusive Events. For example, in tossing a coin the event head and tail are mutually exclusive, because if the outcome is head, then the possibility of getting a tail in the same trial is ruled out. Probability Dr. Manjula Gunarathna 29

30 Addition theorem or theorem on total probability If n events are mutually exclusive, then the probability of happening of any one of them is equal to the sum of probabilities of the happening of the separate events. P(A or B) = P (A) + P (B) P (A B) = P(A) + P (B) Probability Dr. Manjula Gunarathna 30

31 Addition theorem for compatible events The probability of the occurrence of at least one of the events A and B (not mutually exclusive) is given by P ( A B) = P (A) + P (B) P (A B) P ( A or B) = P (A) + P (B) P (A and B) A B Probability Dr. Manjula Gunarathna 31

32 Independents events If two events say A and B are independent, then P (A and B) = P (A). P (B) P (A B) = P (A). P (B) Probability Dr. Manjula Gunarathna 32

33 Complementary events The events A occurs and the event A does not occur are called complementary events. P (A) + P (A) = 1 P (A) = 1 P (A) Probability Dr. Manjula Gunarathna 33

34 (Conditional Probability) The probability of the happening of an event B, when it is known that A has already happened, is called the conditional probability of B and is denoted by p (B/A) P (B/A) = P (A B) / P (A) Probability Dr. Manjula Gunarathna 34

35 Factorial n Factorial n is the continued product of first n natural numbers. Factorial n is symbolically written as n! n! = n (n-1) (n -2) Probability Dr. Manjula Gunarathna 35

36 by definition 0!=1 5! = = 120 4! 3! = ( ) (3.2.1) = 144 n! = n (n-1)! 5! = 5.4! = 120 8! = ! = = ! 3! Probability Dr. Manjula Gunarathna 36

37 A Permutation is an arrangement of items in a particular order. To find the number of Permutations of n items chosen r at a time, you can use the formula n p r ( n n! r)! where 0 r n. Probability Dr. Manjula Gunarathna 37

38 A Combination is an arrangement of items in which order does not matter. Since the order does not matter in combinations, there are fewer combinations than permutations. The combinations are a "subset" of the permutations. To find the number of Combinations of n items chosen r at a time, you can use the formula n! C n r r!( n r)! where 0 r n. Probability Dr. Manjula Gunarathna 38

39 Mathematical expectation or expected values Mathematical expectation of a random variable is obtained by multiplying each probable value of the variable by its corresponding probability and then adding these products. Probability Dr. Manjula Gunarathna 39

40 X E (x) = x E(X)= n Xi P(xi) i=1 Probability Dr. Manjula Gunarathna 40

41 Variance V(x) = var (x) = E(x 2 ) [E(x)] 2 Probability Dr. Manjula Gunarathna 41

42 n E(X 2 ) = Xi 2 P(xi) i=1 Probability Dr. Manjula Gunarathna 42

43 Theorems on Mathematical Expectation 1. Expected value of constant term is constant, that is, if C is constant, then E(C)= C 2. If C is constant, then E(CX) = C.E(X) 3. If A and B are constants, then E(aX ± b)= a.e(x) ± b Probability Dr. Manjula Gunarathna 43

44 4. If a, b and c are constants, then E {(ax+b)/c} = 1/c {a E(x) + b} 5. If X and Y are two random variables, then E(X+Y) = E(X)+E(Y) 6. If X and Y are two independent random variables, then E(XY) = E(X).E(Y) Probability Dr. Manjula Gunarathna 44

45 Theorems on variance of a random variable 1. If c is constant then, V(CX) = C 2 V(X) 2. Variance of constant is zero V(C) = 0 3. If X is a random variable and C is a constant then, V(X+C)= V(X) Probability Dr. Manjula Gunarathna 45

46 4. If a and b are constants then V(aX+b) = a 2 V(X) 5. If X and Y are two independent random variables, then V(X+Y) = V(X)+V(Y) V(X-Y) = V(X)+V(Y) Probability Dr. Manjula Gunarathna 46

47 Bayes Theorem (Inverse Probability Theorem) British mathematician thomas bayes ( ) Probability Dr. Manjula Gunarathna 47

48 Let A 1, A 2.A k be the set of n mutually exclusive and exhaustive events whose union is the random sample space S, of an experiment. If B be any arbitrary event of the sample space of the above experiment with P(B) ǂ 0, then the probability of event A k, when the event B has actually occurred is given by P(A k /B), where P(A k /B)= P(B A k )= P(B/A k )P(A k ) p(b) {P(B/A 1 )P(A 1 )+ P(B/A 2 )P(A 2 ) P(B/A k )P(A k } Probability Dr. Manjula Gunarathna 48

49 The binomial distribution describes the behavior of a count variable X if the following conditions apply: 1: The number of observations n is fixed. 2: Each observation is independent. 3: Each observation represents one of two outcomes ("success" or "failure"). 4: The probability of "success" p is the same for each outcome. Probability Dr. Manjula Gunarathna 49

50 1.probability of success (p) = p 2.probability of failure = q 3.number of trials = n 4.number of successes out of those trials = x probability distribution function P(x) = n C x p x (q) n-x x = 0,1,2... n Probability Dr. Manjula Gunarathna 50

51 Mean value of X: μ = np Variance of X: σ 2 = np(1-p) Standard Deviation of X: σ = (np(1-p)) Probability Dr. Manjula Gunarathna 51

52 Normal Distribution The basic parameters of the normal distribution are as follows: Mean = median = mode = µ Standard deviation = σ Skewness = kurtosis = 0 Probability Dr. Manjula Gunarathna 52

53 The area under the curve in the interval μ σ < x < μ + σ is approximately 68.26% of the total area under the curve. The area under the curve in the interval μ 2σ < x < μ + 2σ is approximately 95.44% of the total area under the curve and the area under the curve in the interval μ 3σ < x < μ + 3σ is approximately 99.74% of the area under the curve. Probability Dr. Manjula Gunarathna 53

54 Probability Dr. Manjula Gunarathna 54

55 If X and Y are independent random variables that are normally distributed (and therefore also jointly so), then their sum is also normally distributed. i.e., if X N (µ 1, σ 2 1) Y N (µ 2, σ 2 2 ) Then X+Y N (µ, σ 2 ) µ=µ 1 +µ 2 σ 2 = σ 2 1+ σ 2 2 σ= σ 2 1+ σ 2 2 Probability Dr. Manjula Gunarathna 55