BERNOULLI TRIALS and RELATED PROBABILITY DISTRIBUTIONS

Transcription

1 BERNOULLI TRIALS and RELATED PROBABILITY DISTRIBUTIONS A BERNOULLI TRIALS Consider tossing a coin several times It is generally agreed that the following aly here ) Each time the coin is tossed there are two ossible outcomes that can be observed: Heads and Tails ) At each toss of the coin, the robability of observing Heads is the same, namely onehalf That is, there is a constant robability of observing Heads ) What haens (ie Heads or Tails) on any one toss of the coin does not affect what haens on another toss That is, outcomes at successive tosses are indeendent of each other Reeated trials of a simle eeriment having these three roerties are called Bernoulli Trials, and an eeriment of this tye is called a Bernoulli Eeriment Definition: Bernoulli Trials are reeated trials of a simle eeriment having the following three roerties () At each trial there are two ossible outcomes, commonly referred to as Success (the event of interest or being studied occurs) and Failure (the event does not occur) () The robability of a Success is constant at each trial - and hence so is the robability of Failure The robability of Success is usually denoted and that of Failure q - () Outcomes at successive trials are indeendent of one another Eamle : Suose a coin is tossed several times What is the robability that Heads is observed for the first time on the fourth toss of the coin? Let H denote Heads (Success) and T denote Tails (Failure) Then / and q / In order for Heads (Success) to occur for the first time on the fourth toss, the sequence of outcomes must have the form TTTH or FFFS [Note: We don't need to worry about outcomes at the fifth and later tosses of the coin since they do not imact on the stated event] The robability of the sequence TTTH is determined as follows P[ TTTH ] P[ T T T H ] in which the subscrit refers to the trial number Because of the indeendence assumtion (roerty ()) which tells us that, if A and B are indeendent events then P [ A B] P[ A] P[ B] (with this multilication roerty being etendable to more than two events), this becomes P[ TTTH ] P[ T T T H ] P[ T ] P[ T ] P[ T ] P[ H ] Proerty () of constant robabilities then allows us to obtain P[ TTTH ] P[ T T T H ] P T P T P T P H [ ] [ ] [ ] [ ] qqq Page of

2 Eamle : Suose a die is rolled several times What is the robability that a "multile of " is observed for the first time on the fourth roll? This is similar to Eamle Let S denote the event "a multile of is observed" so that F is the event "a multile of is NOT observed" The multiles of that are ossible when rolling a die are "" and "" Hence P[ S] and q P[ F] The required robability is then determined to be P[ FFFS] P[ F F F S ] P F P F P F P S [ ] [ ] [ ] [ ] qqq 8 8 B Situations resulting in Bernoulli trials Bernoulli trials are considered to eist in the following situations a) In situations like tossing a coin or rolling a die in which the number of ossible outcomes is obviously fied from trial to trial (eg the numbers on a die do not disaear once seen) One might construct dice that have different numbers of sides For eamle, a yramid built with equilateral triangles is like a -sided die b) If items are drawn one-after-another with relacement from a grou of N items, and if M of the items have a certain roerty, then Success can be defined as obtaining an item with the certain roerty and its robability is then M N c) In the revious case, if items are selected one-after-another without relacement but relatively few are drawn in comarison to the sizes of N, M and N-M, then it is usually assumed that the Bernoulli trials situation is a very good aroimate model [See the section on the Hyergeometric Distribution for the case in which the number drawn is not considered to be small relative to these values] A useable guideline is to require that the number drawn n be not more than % of either M or N-M d) If information is rovided about the roortion or ercentage (ie or %) of the elements of a oulation of infinite size that have a certain roerty, then samling from such a oulation is effectively the same as samling with relacement e) If information is rovided about the roortion or ercentage (ie or %) of the elements of a very large oulation that have a certain roerty but no mention is made about the actual size of the oulation, assume that the oulation is infinite as in the revious case Eamle : Suose that, within the adult oulation of Canada (say of size N,,), 7% of the adult citizens believe that governments are inefficient in their handling of taayers' money Thus n,, have the roerty of having this oinion If three eole are selected at random and without relacement one-after-another from this oulation, what is the robability that only the second selected erson believes this inefficiency idea? Let S denote the event that a selected erson believes this inefficiency idea The sequence of selections must then result in FSF and the robability required here is P[FSF] If it is assumed that the Bernoulli Trials model rovides a good aroimation, then 7 and q and Page of

3 P [ FSF] P[ F S F ] P[ F ] P[ S ] P[ F ] ( )( 7)( ) ( ) ( 7) How close to the correct answer is the aroimation obtained assuming a Bernoulli Trials model? Noting that samling is actually done without relacement, the effective oulation size changes after each selection and the number of elements having the certain roerty may also change after a selection Hence conditional robabilities are required The solution is then obtained as P[ FSF] P[ F S F ] P[ F ] P[ S F ] P[ F ( F S )],,,,,999,999,, 9,999,999 9,999, so that the Bernoulli Trials solution gives a value very close to this correct value suggesting that it is a useful aroimate method When can one assume that the Binomial Trials solution (which effectively assumes samling is done with relacement) rovides an adequate aroimation to the correct value based on samling without relacement? A reasonable guideline to follow (or "rule-of-thumb") is that the aroimation is adequate if both the oulation size N, the number having the roerty M and the number not having the roerty N-M are all "large" with the number of trials or selections n being not more than % of M or N-M In Eamle above, N,,, M,,, N - M,, and the number n selected is a very tiny ercentage of each of these values Eercises: () A die is rolled reeatedly until the number "" is first seen What is the robability that eactly 8 rolls are required? [Answer: 8] () A die is rolled reeatedly until a "multile of " is seen eactly twice What is the robability that eactly four rolls are required? [Answer: 888] () Fifty-five ercent of the students registered at a very large Canadian university are female If four students are selected one-after-another at random and without relacement from the student body of this university, what is the robability that the sees of the selected students will alternate between male and female (or female and male)? [Answer: ] C BINOMIAL EXPERIMENTS and the BINOMIAL DISTRIBUTION A Binomial eeriment is a Bernoulli Trials eeriment in which the number of trials is fied in advance For eamle, a coin is tossed times, or a four dice are rolled simultaneously, or a Page of

4 samle of, eole is taken from the adult oulation of Canada The number of trials (here, and resectively) are known/fied in advance A Binomial Eeriment has the following four roerties () There are a fied number n of reeated trials of a simle eeriment () At each trial there are two ossible outcomes, commonly referred to as Success (the event of interest or being studied occurs) and Failure (the event does not occur) () The robability of a Success is constant at each trial - and hence so is the robability of Failure The robability of Success is usually denoted and that of Failure q - () Outcomes at successive trials are indeendent of one another When a Binomial eeriment is erformed, the usual random variable of interest is "the number of Successes observed in the n-trial Binomial eeriment" If X denotes such a random variable, how are robabilities of events such as X found in such a case? What is the robability distribution of the random variable X here? The following two eamles will hel in understanding the result Eamle : A coin is tossed times What is the robability that eactly Heads are observed in the tosses? Let a random variable be defined as X "the number of Heads observed in tosses of a coin" Then what is P[X ]? One way in which tosses of a coin can result in eactly Heads (ie Successes) is if the sequence HHTTH occurs (note that this is just one ossible sequence) What is the robability of this articular sequence? The sequence can be considered as the outcome of five Bernoulli trials in which q / and its robability found as in eamles, and Hence P [ HHTTH ] P[ H H T T H ] P[ H ] P[ H ] P[ T ] P[ T ] P[ H ] qq But there are other sequences that also lead to eactly Heads For eamle, TTHHH and HTHTH are two others Note that each of these sequences each involves three H's and two T's and has eactly the same robability as our first sequence, namely / The required robability P[ Heads in tosses] P[X ] is thus the sum of the robability / as many times as there are different sequences involving Heads and Tails A comlete listing of the ossible sequences is as follows HHHTT HHTHT HHTTH HTHHT HTHTH HTTHH THHHT THHTH THTHH TTHHH Each of these sequences has the robability / so P[X ] /, the sum of / ten times Is there a systematic way of counting the number of sequences each of which has the same robability? The answer is yes and the number is determined as follows The sequence of five Bernoulli trials is to involve eactly Successes Each way that of the five trials can be selected so as to be the trials at which Success occurs yields a different sequence The number of Page of

5 ways that trials (or objects) can be selected from trials (or objects) is C Thus the answer to this roblem could be found as PHs [ ' in trials ] One simlicity in this eamle is that q / The net eamle is more tyical in that the values of and q are not the same Eamle : Four dice are rolled simultaneously What is the robability that eactly two "multiles of " are observed on the four dice? If a random variable is defined as Y "the number of "multiles of " observed on four dice", then what is P[Y ]? One sequence that gives eactly two "multiles of " is SFSF, where Success denotes obtaining a "multile of " The robability of this sequence is [ ] P[ S F S F ] P[ S ] P[ F ] P[ S ] P[ F ] P SFSF qq Adoting the counting aroach in eamle, the number of different sequences each having two Successes equals the number of ways that trials can be selected from the Bernoulli trials; that is C Hence P[Y ] /8 /8 8/7 Alternatively, 8 [ ] P Y 8 The BINOMIAL DISTRIBUTION Let X be a random variable counting the number of Successes observed in a Binomial eeriment having n trials with a constant robability of Success The ossibilities for the number of successes observed are,,,, n so that the value sace for this random variable is V X {,,,,n} and the robability mass function for X is n ( ) [ ] n X P X q () for each VX {,,,, n} A short-hand indication of this robability distribution is X ~ B n, ( ) Page of

6 Comments: () The arameters of this distribution are n and Knowing them and that ( ) n, B ~ X is enough to be able to roceed with robability calculations () The eression () is obtainable as follows In order for the random variable X to have the articular value, the n Bernoulli trials must result in eactly Successes and hence eactly n - Failures The robability of any one such sequence is then The number of different sequences of n trials that have Successes and n - Failures equals the number of ways of selecting of the n trials to be "Success" trials with the remaining n - being "Failure" trials, that is Each of these has the robability so that q n [ ] X P n q n ( ) n X q n Eamle : Suose a die is rolled times and Success is defined as obtaining a "multile of " If X counts the number of successes in the rolls, then ( B, ~ X ) The robability distribution of this random variable can be resented as in the following table Possible value: Probability Formula: ( ) [ ] X X P Calculated Probability Total Page of

7 Eamle 7: A rofessional "darts" layer successfully hits the region of the dart board at which he aims with constant robability of 9 Because of his eerience in laying the game, it is reasonable to assume that his results on successive throws (ie hitting the region at which he aims, or not) are indeendent events What is the robability that in his net nine throws he hits the region at which he is aiming eactly si times? What is the robability that in these nine throws he hits his targeted region at least eight times? Let U be a random variable counting the number of times the dart layer hits the region at which he is aiming in nine throws of the darts Then U~B(9,9) (i) What is the robability that in his net eight throws he hits the region at which he is aiming eactly si times? 9 P[ U ] ( 9) ( ) (ii) What is the robability that in these nine throws he hits his targeted region at least eight times? 9 9 u 9u P[ U 8] ( 9) ( ) u 8u ( 9) ( ) + ( 9) ( ) Eamle 8: Recent reorts suggest that only % of Canadians are for changes to the medicare system that would allow rivately run hositals A random samle of Canadians is obtained Find the robabilities that, for this random samle, (i) eactly are for such changes; (ii) at most are for such changes; (iii) more than two are for such changes Let V be a random variable that counts the number in the samle of size who are for such changes Then V~B(,) (i) eactly are for such changes P V [ ] ( ) ( 8) 8 (ii) at most are for such changes v v P[ V ] ( ) ( 8) v v ( ) ( ) ( ) ( ) ( ) ( 8) (iii) more than two are for such changes P V > P V 8 [ ] [ ] 7 Page 7 of

8 Mean and Standard Deviation of a Binomial Distribution If X is a Binomially distributed random variable with arameters n and, that is X ~ B( n, ), then its mean (eected value), variance and standard deviations are defined resectively as n n n µ E[ X ] ( ) q n, σ n n [ ] ( µ ) ( ) ( ) [ X ] E ( X µ ) Var µ q n [ X ] σ σ SD These are rather comlicated looking eressions but can be evaluated in given situations The following table outlines how these calculations can be done in table form for the random variable given in Eamle [Comment: The calculations in the column headed "( ) ( ) µ " are easier in this eamle than they usually are since the mean µ has an integer value] n Value: Probability: () ( ) ( µ ) ( ) Total ( ) ( ) 79 9 and ( ) 79 ( ) 79 ( ) 79 ( ) 79 ( ) µ σ 79 Fortunately it is not necessary to go through this amount of work The following theorem indicates how the mean, variance and standard deviation can always be obtained when dealing X ~ B n, with a Binomial random variable ( ) Page 8 of

9 Theorem : Suose X is a Binomial random variable with arameters n and ; that is, ~ B( n, ) Then the mean or eected value of X is µ n, the variance of X is σ nq, and the standard deviation of X is σ nq X Alying these results in the revious eamle gives µ n σ nq σ nq 7 Comment: ABinomial random variable X counts the number of successes in n trials Hence the mean or eected value may be referred to as the eected number of successes For eamle, what is the eected number of Heads if a coin is tossed 9 times? The number of Heads in 9 tosses of a coin is a B(9,) random variable, so the eected number of Heads is µ E [ X ] n 9 The eected number of Heads is the mean number of Heads to be eected when a fair coin is tossed 9 times Remember that the mean or eected number need not be a ossible number (as in this case in which the eected number is not a number of Heads that would ever be achieved in 9 tosses of a coin) Eercises : () In attemting free throws during basketball games, a certain layer has a constant robability of 8 of making the shot Furthermore his successes (or lack of same) from shot to shot are indeendent events What is the robability that he makes eactly three-quarters of the free throws he attemts one game? What is the eected number of free throws that he would make in these attemts? What is the standard deviation of the number of successful free throws in attemts? () An urn contains red balls and blue balls Balls are drawn one after another with relacement from this urn Suose a total of 7 draws are made What is the robability that red balls are drawn eactly times? What is the robability that red balls are drawn at most once? What is the robability that red balls are drawn at least twice? What is the eected number of red balls in the 7 draws? B The GEOMETRIC DISTRIBUTION A Binomial eeriment involves a fied number n of Bernoulli trials Another interesting eeriment is to erform Bernoulli trials reeatedly until a "Success" is observed for the first time The question of interest is: how many rolls are required for this to haen? Eamle 9: A die is rolled reeatedly until the first time that a "multile of " (a Success) is observed How many rolls are required? What is the robability that eactly five rolls are Page 9 of

10 required? In these Bernoulli trials, / is the robability of observing a "multile of " at any articular roll If the first Success occurs on the fifth roll, then each of the first four rolls must have resulted in Failure Thus the sequence of outcomes would be FFFFS The robability of this sequence is just P[ FFFFS] P[ F F F F S ] P F P F P F P F P S [ ] [ ] [ ] [ ] [ ] qqqq See eamles, and for similar Bernoulli trials solutions In the above eamle, a random variable Y could be defined by "Y the roll number on which a multile of is first obtained" The robability calculated here is then Y ( ) P[ Y ] q This random variable Y is said to have a Geometric distribution with single arameter / Its value sace is the set of all ossible values for Y, namely all ossible numbers of trials that could be required in order to obtain a Success for the first time Thus V Y {,,, }, the set of all ositive integers The number of trials required could be any ositive integer without uer bound It is conceivable (but unlikely) to continue to roll the die reeatedly without ever observing a Success, and hence the value sace has no uer limit If y V Y {,,, }, what is Y ( y) P[ Y y]? In order for the y th Bernoulli trial to be the trial at which the first Success occurs, all y - revious trials must result in Failure Thus the sequence of outcomes yielding this is FFF FS in which there are y - F's followed by one S Thus ( y) P[ Y y] q y Y for y VY,,, { } Definition: A random variable Y is said to have a Geometric distribution with arameter - that is, Y Ge - if its value sace is V,,, and its robability function is given by ~ ( ) Y { } ( y) P[ Y y] q y Y V {,,, } for y Y Comment: Since the eeriment is to wait until the first Success occurs, this distribution might be referred to as a waiting time distribution Eamle : A student writing a multile choice test is faced with answer choices for each question If he randomly selects an answer for each question (that is, he guesses), what is the robability that the first correct answer is obtained on the seventh question? Page of

11 Let Y be a random variable counting the number of guesses until he finally gets one right Assuming his guesses are at random and done indeendently from question to question, it follows that Y Ge( / and 7 P Y 7 9 7,,, ~ ) () [ ] Y Eamle : Seventy ercent of students hold jobs while attending university Suose students are selected one-after-another at random from a large class What is the robability that the first student selected who does NOT hold a job this year is the fourth student selected? The question asks about finding a student who does NOT hold a job Thus Success is defined as finding such a student Selecting students at random will be assumed to give us the "indeendence" of outcomes at various trials (selections), and the robability of a success at each W ~ Ge, the required robability is trial is (7% hold jobs so % do not) Letting ( ) ( ) P[ W ] ( 7) ( ) 9 W How long should one eect to have to wait until the first success is observed? That is, what is the mean or eected value of a Geometric random variable? Theorem : Suose Y is a Geometric random variable with arameter ; that is, Y the mean or eected value of Y is µ, the variance of Y is σ q, and q the standard deviation of Y is σ ~ Ge( ) Then Eamle (continued): How many students would one eect to have to select until one who does not work while attending university is found? If this rocedure is reeated for different classes, what is the standard deviation of the numbers of students, from class to class, who would have to be selected in order to find one who does not work? The mean or eected number is µ and the standard deviation is σ q 7 ( ) Thus, on average, students would have to be selected in order to find one who does not work, but there is a relatively large amount of variability from attemt to attemt as measured by the standard deviation of 7889 (which is 87% of the mean) Comments: Page of

12 () The definition of a robability distribution requires that the robability function satisfy the two roerties a) Y y for every y VY and b) ( y) In the Y ( y) ( ) Y all y V Y ( y) q case of the Geometric distribution, the first of these is clearly satisfied since Y is greater than zero since both of and q are Eamining the second roerty roduces the following summation Y all y V Y ( y) q y y q + q + q ( + q + q + q + ) + q + In order for this sum to equal, the sum q + q + q + must equal / for then the above would become ( + q + q + q + ) as required Since - q, the ratio / is equivalent to and thus + q + q + q + This latter eression is called a q q "series eansion" for /(-q) + y Eercises : () Consider the "darts" layer in Eamle 7 What is the eected number of throws until he first misses the region at which he aims? What is the robability that he first misses on his sith throw? What is the robability that his first miss occurs on his third or fourth throw? What is the robability that his first miss comes later than his fifth throw? () Consider the roblem of samling adult Canadians regarding their oinions on the issue of government inefficiency in the handling of taayers' money as in Eamle What is the eected number of adult Canadians samled until one is found who agrees that governments are inefficient in handling taayers' money? What is the eected number of adult Canadians samled until one is found who does not agree that governments are inefficient in handling taayers' money? What is the robability that the first erson who agrees that governments are inefficient in handling taayers' money is the third erson samled? () Suose a telehone banking system is busy % of the time That is, the robability that a erson finds the service busy when honing in is What is the robability that you finally get through to the banking system on your fourth attemt? What is the mean number of attemts that are needed in order to finally get through to the banking system? C The NEGATIVE BINOMIAL DISTRIBUTION The Geometric distribution is the aroriate robability distribution in a sequence of Bernoulli trials when waiting for the first Success to occur What is the robability distribution in a sequence of Bernoulli trials when waiting for two successes, or three successes, etc? In general, what is the waiting time distribution to the k th success in a sequence of Bernoulli trials, for k,,,? The answer is the Negative Binomial distribution Suose that 8% of the automobile engines roduced on one assembly line are defective If engines are selected at random from the assembly line, what is the robability that the second Page of

13 defective engine is the fourth engine selected? Letting D denote defective and N denote nondefective, the different sequences that lead to finding a second defective as the fourth engine selected are DNND and NDND and NNDD Note the following two roerties of each of these sequences: the fourth engine is Defective, and there is one Defective engine amongst the first three selected The robability of a Defective engine is 8 and of a non-defective engine 9 Thus the robability of each of the three sequences is q ( 9) ( 8) 9 the required robability is q ( 9) ( 8) 88 and How can the event "second defective as fourth engine selected" occur? The answer is: there must be eactly one defective in the first three selected, and the fourth one must then be another defective What is the robability of eactly one defective in the first three selected? This is just a Binomial distribution question, say Y~B(,8), and P[ Y ] ( 8) ( 9) Multilying this answer by 8 to account for the second defective at the fourth draw gives the required answer as above This is the essential nature of the Negative Binomial distribution In waiting for the k th success in a sequence of Bernoulli trials with arameter, eactly trials will be required if the first - trials result in k - successes, and then the th trial results in the k th success Note that at least k trials will be required to achieve k successes so that the value sace of the random variable is { k,k +,k +, } Definition: A random variable Y is said to have a Negative Binomial distribution with arameters k and - that is, X ~ NB( k, ) - if its value sace is V X { k,k +,k +, } and its robability function is given by ( ) [ ] k ( ) ( k ) k k X P X q q k k VX k,k +,k +, for { } In waiting for the k th success, it is necessary to first wait for the first success, then for the second success, and so on, and then for the k th success This results in the mean and variance of a Negative Binomial random variable each being k times the mean and variance of a Geometric random variable with the same robability arameter Theorem : Suose X is a Negative Binomial random variable with arameters k and ; that is, X ~ NB( k, ) Then the mean or eected value of X is µ k, the variance of X is σ kq, and Page of

14 kq the standard deviation of X is σ Eamle : A wildcat oil well drilling comany successfully strikes oil on % of the wells it drills The comany has a new field to elore and begins drilling wells one after another What is the robability that the third successful well will be the eighth one drilled? What is the eected number of wells that must be drilled in order to find oil three times? A Negative Binomial random variable with k and is called for Let X ~ NB(, ) The robability that the third successful well is the eighth one drilled is X ( 8) P[ X 8] ( ) ( 8) ( ) ( 8) 8 78 The eected number of wells to drill in order to find a third successful one is k µ [Note: This eected value is reasonable The comany is successful on one-fifth of its attemts suggesting an average of one success in every five attemts Thus attemts would be required - on average - to obtain three successes] Eamle : Suose that female children are born for every 8 male children A coule decides that it wants to have two sons What is the robability that, if they kee having children until they have their second son, they will end u with five children? Assume the sees of the children are indeendent of each other What is the eected number of children in such a family wanting to have two sons? Let Y be a random variable counting the number of children in this family Since the family will sto having children once a second son is born, Y is a Negative Binomial random variable with arameters k and 8/(+8) 8 That is, Y ~ NB(,8) The robability that the family has five children is P[ Y ] ( 8) ( ) ( 8) ( ) 98 k The eected number of children in the family is µ 7 8 Eercises : () A die is rolled until a "" is observed for the third time What is the robability that eactly seven rolls are required? What is the robability that at most four rolls are required? What is the eected number of rolls required? Page of

15 () In the NHL's Stanley Cu Playoffs, two teams lay each other until one team has won four games Suose in one series team A has a robability of of winning any given game, and that the outcomes from game to game are indeendent events What is the robability that team A wins the series in eactly games? What is the robability that team A wins the series in less than games? What is the robability that team A wins the series? Final Comments: () Probability distributions are used as models in finding robabilities of events Each model is built uon certain conditions which should hold in order to use that model Sometimes a model will be used even when the conditions are not eactly satisfied but are aroimately so () Remember that the Binomial, Geometric and Negative Binomial distributions are all built uon Bernoulli trials What are the three roerties of Bernoulli trials? What kinds of situations lead to Bernoulli trials? Additional Eercises: () The University of Saskatchewan reorted student enrollment of 9,77 students in the Fall of Of this total, 8% are full-time students while the other % are art-time students Suose students are randomly selected from this large grou a) If students are selected at random, what is the robability that more than two of them are full-time students? How many students in this samle would be eected to be fulltime? b) If students are selected one after another, what is the robability that a art-time student will be selected for the first time as the seventh student selected? How many students would one eect to have to select in order to obtain a art-time student? c) Suose students are selected until a third art-time student is selected What is the robability that students will be selected? How many students would one eect to have to select in order to obtain three art-time students () A major league baseball layer has a batting average This means that he has successfully hit on % of his times at-bat (in the ast) Suose that he enters one game with this batting average as his robability of getting a hit the first time he comes to bat As the game rogresses, his robability of getting a hit changes as follows If he does get a hit during one at-bat, his robability of getting a hit the net time u increases by If he does not get a hit during one at-bat, his robability of getting a hit the net time u decreases by Suose this layer comes to bat five times during one game a) What is the robability that he gets a hit all five times he is at bat? b) What is the robability that he gets no hits during this game? c) What is the robability that he gets eactly one hit during this game? Page of