UNIT 2 DIFFERENT APPROACHES TO PROBABILITY THEORY

UNIT 2 DIFFERENT APPROACHES TO PROBABILITY THEORY Structure 2.1 Itroductio Objectives 2.2 Relative Frequecy Approach ad Statistical Probability 2. Problems Based o Relative Frequecy 2.4 Subjective Approach to Probability 2.5 Axiomatic Approach to Probability 2.6 Some Results usig Probability Fuctio 2.7 Summary 2.8 Solutios/Aswers Differet Approaches to Probability Theory 2.1 INTRODUCTION I the previous uit, we have defied the classical probability. There are some restrictios i order to use it such as the outcomes must be equally likely ad fiite. There are may situatios where such coditios are ot satisfied ad hece classical defiitio caot be applied. I such a situatio, we eed some other approaches to compute the probabilities. Thus, i this uit, we will discuss differet approaches to evaluate the probability of a give situatio based o past experiece or ow experiece or based o observed data. Actually classical defiitio is based o the theoretical assumptios ad i this uit, our approach to evaluate the probability of a evet is differet from theoretical assumptios ad will put you i a positio to aswer those questios related to probability where classical defiitio does ot work. The uit discusses the relative frequecy (statistical or empirical probability) ad the subjective approaches to probability. These approaches, however, share the same basic axioms which provide us with the uified approach to probability kow as axiomatic approach. So, the axiomatic approach will also be discussed i the uit. Objectives After studyig this uit, you should be able to: explai the relative frequecy approach ad statistical(or empirical) probability; discuss subjective approach to probability; ad discuss axiomatic approach to probability. 29

Basic Cocepts i Probability 2.2 RELATIVE FREQUENCY APPROACH AND STATISTICAL PROBABILITY Classical defiitio of probability fails if i) the possible outcomes of the radom experimet are ot equally likely or/ad ii) the umber of exhaustive cases is ifiite. I such cases, we obtai the probability by observig the data. This approach to probability is called the relative frequecy approach ad it defies the statistical probability. Before defiig the statistical probability, let us cosider the followig example: Followig table gives a distributio of daily salary of some employees: Salary per day (I Rs) Below 1 1-15 15-2 2 ad above Employees 2 4 5 15 If a idividual is selected at radom from the above group of employees ad we are iterested i fidig the probability that his/her salary was uder Rs. 15, the as the umber of employees havig salary less tha Rs 15 is 2 + 4 = 6 ad the total umber employees is 2 + 4 + 5 + 15 = 125, therefore the relative frequecy that the employee gets salary less tha Rs. 15 is 125 = 12 25. This relative frequecy is othig but the probability that the idividual selected is gettig the salary less tha Rs. 15. So, i geeral, if X is a variable havig the values x 1, x 2,, x with frequecies f 1, f 2,, f, respectively. The f f f,,..., f f f 1 2 i i i i=1 i=1 i=1 are the relative frequecies of x 1, x 2,, x respectively ad hece the probabilities of X takig the values x 1, x,, x respectively. But, i the above example the probability has bee obtaied usig the similar cocept as that of classical probability. Now, let us cosider a situatio where a perso is admiistered a sleepig pill ad we are iterested i fidig the probability that the pill puts the perso to sleep i 2 miutes. Here, we caot say that the pill will be equally effective for all persos ad hece we caot apply classical defiitio here. To fid the required probability i this case, we should either have the past data or i the absece of the past data, we have to udertake a experimet where we admiister the pill o a group of persos to check the effect. Let m

be the umber of persos to whom the pill put to sleep i 2 miutes ad be the total umber of persos who were admiistered the pill. Differet Approaches to Probability Theory The, the relative frequecy ad hece the probability that a particular perso will put to sleep i 2 miutes is m. But, this measure will serve as probability oly if the total umber of trials i the experimet is very large. I the relative frequecy approach, as the probability is obtaied by repetitive empirical observatios, it is kow as statistical or empirical probability. Statistical (or Empirical) Probability If a evet A (say) happes m times i trials of a experimet which is performed repeatedly uder essetially homogeeous ad idetical coditios (e.g. if we perform a experimet of tossig a coi i a room, the it must be performed i the same room ad all other coditios for tossig the coi should also be idetical ad homogeeous i all the tosses), the the probability of happeig A is defied as: m P(A) = lim. As a illustratio, we tossed a coi 2 times ad observed the umber of heads. After each toss, proportio of heads i.e. m was obtaied, where m is the umber of heads ad is the umber of tosses as show i the followig table (Table 2.1): Table 2.1: Table Showig Number of Tosses ad Proportio of Heads (Number of Tosses) m (Number of Heads) Proportio of Heads i.e.p(h)=m/ 1 1 1 2 2 1 2.666667 4.75 5 4.8 6 4.666667 7 4.571429 8 5.625 9 6.666667 1 6.6 15 1.666667 2 12.6 25 14.56 16.5 5 18.514286 1

Basic Cocepts i Probability 4 22.55 45 25.555556 5 29.58 6.55 7 41.585714 8 46.575 9 52.577778 1 5.5 12 66.55 14 72.514286 16 82.5125 18 92.511111 2 15.525 The a graph was plotted takig umber of tosses () o x-axis ad proportio of heads m o y-axis as show i Fig. 2.1. Number of tosses () Fig. 2.1: Proportio of Heads versus Number of Tosses The Graph reveals that as we go o icreasig, 2 m teds to 1 2

m 1 i.e. lim = 2 Hece, by the statistical (or empirical) defiitio of probability, the probability of gettig head is m 1 lim =. 2 Statistical probability has the followig limitatios: (i) The experimetal coditio may get altered if it is repeated a large umber of times. (ii) m lim may ot have a uique value, however large may be. Differet Approaches to Probability Theory 2. PROBLEMS BASED ON RELATIVE FREQUENCY Example 1: The followig data relate to 1 couples Age of wife 1-2 2- -4 4-5 5-6 Age of Husbad 15-25 6 25-5 16 1 5-45 1 15 7 45-55 7 1 4 55-65 4 5 (i) (ii) Fid the probability of a couple selected at radom has a age of wife i the iterval 2-5. What is the probability that the age of wife is i the iterval 2-4 ad the age of husbad is i the iterval 5-45 if a couple selected at radom? Solutio: (i) Required probability is give by (ii) = (+16+1+ +) +( +1 +15+7 +) +(+ +7 +1+ 4) 1 82 = =.82 1 1+15 25 Required probabilty = =.25 1 1

Basic Cocepts i Probability Example 2: A class has 15 studets whose ages are 14, 17, 15, 21, 19, 2, 16, 18, 2, 17, 14, 17, 16, 19 ad 2 years respectively. Oe studet is chose at radom ad the age of the selected studet is recorded. What is the probability that (i) the age of the selected studet is divisible by, (ii) the age of the selected studet is more tha 16, ad (iii) the selected studet is eligible to pole the vote. Solutio: Age X Frequecy f Relative frequecy 14 15 16 17 18 19 2 21 2 1 2 1 2 1 2/15 1/15 2/15 /15 1/15 2/15 /15 1/15 (i) The age divisible by is 15 or 18 or 21. 111 1 Re quired Pr obability 15 15 5 (ii) Age more tha 16 meas, age may be 17, 18, 19, 2, 21 1 2 1 1 2 Re quired Pr obability 15 15 (iii) I order to poll the vote, age must be 18 years. Thus, we are to obtai the probability that the selected studet has age 18 or 19 or 2 or 21. 1 2 1 7 Re quired Pr obability 15 15 Example : A tyre maufacturig compay kept a record of the distace covered before a tyre eeded to be replaced. The followig table shows the results of 2 cases. Distace (i km) Less tha 4 41-1 11-2 21-4 Frequecy 2 1 2 15 18 More tha 4 4 If a perso buys a tyre of this compay the fid the probability that before the eed of its replacemet, it has covered (i) at least a distace of 41 km.

(ii) at most a distace 2 km (iii) more tha a distace 2 km (iv) a distace betwee 1 to 4 Solutio: The record is based o 2 cases, Exhaustive casesi each case 2 Differet Approaches to Probability Theory (ii) (i) Out of 2 cases, the umber of cases i which tyre covered at least 41 km = 1 + 2 + 15 + 18 = 198 198 198 99 Re quired Pr obability 2 2 1 Number of cases i which distace covered by tyres of this compay is at most 2 km = 2 + 1 + 2 = 2 2 2 4 Re quired Pr obability 2 2 25 (iii) Number of cases i which tyres of this compay covers a distace of more tha 2 = 15 + 18 = 168 168 168 21 Re quired Pr obability 2 2 25 (iv) Number of cases i which tyres of this compay covered a distace betwee 1 to 4 = 2 + 15 = 17 17 17 Re quired Pr obability 2 2 Now, you ca try the followig exercises. E 1) A isurace compay selected 5 drivers from a city at radom i order to fid a relatioship betwee age ad accidets. The followig table shows the results related to these 5 drivers. Age of driver (i years) Accidets i oe year Class iterval 1 2 4 or more 18-25 6 26 185 9 7 25-4 9 24 16 85 65 4-5 1 195 15 7 5 5 ad above 5 17 12 6 If a driver from the city is selected at radom, fid the probability of the followig evets: 5

Basic Cocepts i Probability E 2) E ) (i) Age lyig betwee 18-25 ad meet 2 accidets (ii) Age betwee 25-5 ad meet at least accidets (iii) Age more tha 4 years ad meet at most oe accidet (iv) Havig oe accidet i the year (v) Havig o accidet i the year. Past experiece of 2 cosecutive days speaks that weather forecasts of a statio is 12 times correct. A day is selected at radom of the year, fid the probability that (i) weather forecast o this day is correct (ii) weather forecast o this day is false Throw a die 2 times ad fid the probability of gettig the odd umber usig statistical defiitio of probability. 2.4 SUBJECTIVE APPROACH TO PROBABILITY I this approach, we try to assess the probability from our ow experieces. This approach is applicable i the situatios where the evets do ot occur at all or occur oly oce or caot be performed repeatedly uder the same coditios. Subjective probability is based o oe s judgmet, wisdom, ituitio ad expertise. It is iterpreted as a measure of degree of belief or as the quatified judgmet of a particular idividual. For example, a teacher may express his /her cofidece that the probability for a particular studet gettig first positio i a test is.99 ad that for a particular studet gettig failed i the test is.5. It is based o his persoal belief. You may otice here that sice the assessmet is purely subjective oe, it will vary from perso to perso, depedig o oe s perceptio of the situatio ad past experiece. Eve whe two persos have the same kowledge about the past, their assessmet of probabilities may differ accordig to their persoal prejudices ad biases. 2.5 AXIOMATIC APPROACH TO PROBABILITY All the approaches i.e. classical approach, relative frequecy approach (Statistical/Empirical probability) ad subjective approach share the same basic axioms. These axioms are fudametal to the probability ad provide us with uified approach to probability i.e. axiomatic approach to probability. It defies the probability fuctio as follows: Let S be a sample space for a radom experimet ad A be a evet which is subset of S, the P(A) is called probability fuctio if it satisfies the followig axioms (i) P(A) is real ad P(A) (ii) P (S) = 1 (iii) If A 1, A 2,... is ay fiite or ifiite sequece of disjoit evets i S, the P(A1 or A2 or...or A ) = P(A 1) + P(A 2) +...+ P(A ) 6

Now, let us give some results usig probability fuctio. But before takig up these results, we discuss some statemets with their meaigs i terms of set theory. If A ad B are two evets, the i terms of set theory, we write Differet Approaches to Probability Theory i) At least oe of the evets A or B occurs as A B ii) Both the evets A ad B occurs as A B iii) Neither A or B occurs as A B iv) Evet A occurs ad B does ot occur as A B v) Exactly oe of the evets A or B occurs as (A B) (A B ) vi) Not more tha oe of the evets A or B occurs as A B A B A B. Similarly, you ca write the meaigs i terms of set theory for such statemet i case of three or more evets e.g. i case of three evets A, B ad C, happeig of at least oe of the evets is writte as A B C. 2.6 SOME RESULTS USING PROBABILITY FUNCTION 1 Prove that probability of the impossible evet is zero Proof: Let S be the sample space ad be the set of impossible evet. S S P(S ) P(S) P(S) P( ) P(S) [By axiom (iii)] 1+ P( ) 1 [By axiom (ii)] P( ) 2 Probability of o-happeig of a evet A i.e. complemetary evet A of A is give by P(A) =1 P(A) S Proof: If S is the sample space the A A = S [A ad A are mutually disjoit evets] P(A A ) = P(S) P(A) P(A) = P S [Usig axiom (iii)] P(A) 1 P A. Prove that = 1 [Usig axiom (ii)] A A B A A B (i) P(A B) = P(A) P(A B) (ii) P(A B) = P(B) P(A B) A B A B A B A B 7

Basic Cocepts i Probability Proof If S is the sample space ad A, B S the (i) A = (A P(A) = P((A B ) (A B) B ) (A B) ) = P(A B ) + P(A B) [Usig axiom (iii) as A B ad A B are mutually disjoit] P(A B ) = P(A) P(A B) (ii) B = (A B) (A B) P(B) = P((A B) (A B)) = P(A B) + P(A B) [Usig axiom (iii) as A B ad A B are mutually disjoit] P(A B) = P(B) P(A B) Example 4: A, B ad C are three mutually exclusive ad exhaustive evets associated with a radom experimet. Fid P(A) give that : P B = 4 1 PA ad P C = P B Solutio: As A, B ad C are mutually exclusive ad exhaustive evets, A B C = S P(A B C) = P(S) P(A) + P(B) + P(C) = 1 1 P B =1 P(A) + 4 P(A) + 1 P A + P A =1 4 4 P(A) + 1 1+ + P A =1 4 4 2P(A) =1 u si g axiom (iii) as A,B,C are mutually disjoit evets P(A) = 1 2 8

Examples 5: If two dice are throw, what is the probability that sum is a) greater tha 9, ad b) either 1 or 12. Solutio: a) P[sum > 9] = P[sum = 1 or sum = 11 or sum = 12] = P[sum =1] + P[sum = 11] + P[sum = 12] = + 2 + 1 6 6 6 = 6 = 1 6 6 [usig axiom (iii)] [ for sum = 1, there are three favourable cases (4, 6), (5, 5) ad (6, 4). Similarly for sum =11 ad 12, there are two ad oe favourable cases respectively.] Let A deotes the evet for sum =1 ad B deotes the evet for sum = 12, Re quired probability = P A B P A B [Usig De- Morga's law Differet Approaches to Probability Theory = 1 P(A B) Now, you ca try the followig exercises. (see Uit 1 of Course MST-1)] = 1 [P(A) + P(B)] [Usig axiom (iii)] 1 4 1 8 = 1 + 6 6 1 1 = 6 9 9 E4) If A, B ad C are ay three evets, write dow the expressios i terms of set theory: a) oly A occurs b) A ad B occur but C does ot c) A, B ad C all the three occur d) at least two occur e) exactly two do ot occur f) oe occurs E5) Fourtee balls are serially umbered ad placed i a bag. Fid the probability that a ball is draw bears a umber multiple of or 5. 2.7 SUMMARY Let us summarize the mai topics covered i this uit. 1) Whe classical defiitio fails, we obtai the probability by observig the data. This approach to probability is called the relative frequecy approach ad it defies the statistical probability. If a evet A (say) happes m times i trials of a experimet which is performed repeatedly uder essetially homogeeous ad idetical coditios, the the (Statistical or Empirical) probability of happeig A is defied as 9

Basic Cocepts i Probability P(A) = m lim. 2) Subjective probability is based o oe s judgmet, wisdom, ituitio ad expertise. It is iterpreted as a measure of degree of belief or as the quatified judgmet of particular idividual. ) If S be a sample space for a radom experimet ad A be a evet which is subset of S, the P(A) is called probability fuctio if it satisfies the followig axioms (i) P(A) is real ad P(A) (ii) P (S) = 1 (iii) If A 1, A 2, is ay fiite or ifiite sequece of disjoit evets i S, the P(A1 or A 2or...or A ) = P(A 1) + P(A 2) +...+ P(A ). This is the axiomatic approach to the probability. 2.8 SOLUTIONS/ANSWERS E 1) Sice the iformatio is based o 5 drivers, the umber of exhaustive cases is = 5. Thus, (i) the required probability = (ii) the required probability = (iii) the required probability = (iv) the required probability = (v) the required probability = 185 7 = 5 1 85+65 + 7 +5 5 27 27 = = 5 5 1 +195+5+17 1865 7 = = 5 5 1 26+ 24+195+17 5 6 +9 +1+5 5 = 865 17 = 5 1 = = 5 5 E 2) Sice the iformatio is based o the record of 2 days, so the umber of exhaustive cases i each case = 2. (i) Number of favourable cases for correct forecast = 12 the required probability = 12 12 = = 2 2 5 4

(iii) Number of favourable outcomes for icorrect forecast = 2 12 the required probability = 8 2 = 2 5 = 8 Differet Approaches to Probability Theory E ) First throw a die 2 times ad ote the outcomes. The costruct a table for the umber of throws ad the umber of times the odd umber turs up as show i the followig format: Number of Throws() Number of times the odd umber turs up (m) 1 2... Proportio (m/) 2 Now, plot the graph takig umber of throws () o x-axis ad the proportio ( m ) o y-axis i the maer as show i Fig. 2.1. The see to which value the proportio ( m ) approaches to as becomig large. This limitig value of E 4) a) A B C b) A B C m is the required probability. c) A B C d) A B C A B C A B C A B C e) A B C A B C A B C f) A B C E 5) Let A be the evet that the draw ball bears a umber multiple of ad B be the evet that it bears a umber multiple of 5, the A = {, 6, 9, 12} ad B = {5, 1} 41

Basic Cocepts i Probability P(A) = 4 2 = 14 7 ad P(B) = 2 14 = 1 7 The required probability = P(A or B) = P A + P B = 2 + 1 = 7 7 7 [Usig axiom (iii) as A ad B are mutually disjoit] 42