5.3 Binomial Distributions

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1 5. Biomial Distributios Project Coectio If your project ivolves aalyzig a process i which the results have two possible outcomes oly success or failure the the probability distributio described i this sectio will apply to your aalysis. biomial evet a experimet i which each outcome is the result of Beroulli trials (defied i the box to the right) Thi about? The Value of q Why does q = p? Beroulli trial a idepedet trial that has two possible outcomes: success or failure Parvi Das is a quality-cotrol egieer. Oe of his resposibilities is to moitor the defect rate of a productio lie. Probability distributios based o the results of the Biomial Theorem ca be used as mathematical models to do this. Biomial Experimets Each time a quality-cotrol ispector chooses a item to test, there are two possible outcomes for the test. The item either passes the test ad is used, or it fails ad does ot get used. Each of these outcomes has a probability associated with it. To guaratee cosistecy, the quality-cotrol process is based o certai assumptios. These assumptios form the formal defiitio of a biomial experimet. Biomial Experimet A biomial experimet is ay experimet that has the followig properties:. There are idetical trials. Together, these form a biomial experimet.. The purpose of the experimet is to determie the umber of successes that occurs durig the trials.. There are two possible outcomes for each trial. These are usually termed success ad failure. The probability of a success is usually deoted p ad the probability of a failure is q or p.. The probability of the outcomes remais the same from trial to trial. The values of p ad p do ot chage from oe trial to the ext because the value of p is the same for each trial. 5. The trials are idepedet of oe aother. Repeated trials, which are idepedet ad have two possible outcomes (success ad failure) are called Beroulli trials. They are amed after Daiel Beroulli, a member of a remarable family of mathematicias ad scietists. Daiel s gradfather Nicolaus Sr., his ucles Jacob I ad Nicolaus I, his father Joh I, his brothers Nicolaus III ad Joh II, ad his ephews Joh III ad Jacob II all made sigificat cotributios to mathematics ad sciece. Each member of the Beroulli family was ow for his ee itellect ad fiery temper. After wiig the Frech Academy of Sciece Prize, which his father had usuccessfully tried to wi, Daiel was throw out of the house! 9 CHAPTER 5 PROBABILITY DISTRIBUTIONS AND PREDICTIONS

2 INVESTIGATION: THE DISTRIBUTION OF THE NUMBER OF HEADS IN FOUR COIN TOSSES Suppose you were to simulate samplig a productio lie ow to have a 50% defect rate. You do this by tossig cois. Calculate the probability distributio for selectig four items from this productio lie. The tossig of a coi is a example of a Beroulli trial. A experimet i which each trial requires the toss of a coi several times ad the recordig of the umber of heads is a example of a biomial experimet. Purpose To ivestigate the results of a biomial experimet.? Thi about Step A Why is tossig oe coi four times equivalet to tossig four cois oce? Techoli Usig techology to record your data ad geerate a histogram will save time. biomial radom variable the umber of successes i repeated trials of a biomial experimet biomial distributio the probability distributio of a biomial radom variable Procedure A. Wor with a parter ad record the results of tossig four cois simultaeously. Alteratively, you could create a simulatio of the coi toss usig a spreadsheet, a TI-8 Plus calculator, or Fathom. B. Record the umber of heads. A head will represet a defective item. C. Repeat util you have completed 00 repetitios of the experimet. D. Create a relative frequecy distributio for the umber of heads (defects) i the toss of four cois. E. Either repeat Steps A through C, or combie results with other studets to icrease the umber of trials. F. Create a ew relative frequecy distributio usig the larger data set. Discussio Questios. How did icreasig the umber of repetitios of the experimet affect the shape of the distributio?. Why is tossig a sigle coi oce a trial?. Why is each trial a Beroulli trial?. Why is the umber of heads obtaied i each repetitio a discrete radom variable? THE PROBABILITY DISTRIBUTION FOR A DIE- ROLL SIMULATION I a biomial experimet, the umber of successes i repeated Beroulli trials is a discrete radom variable, usually represeted by the letter X. X is termed a biomial radom variable ad its probability distributio is called a biomial distributio. Rollig a Die Suppose a cereal compay puts oe of six possible prizes i each cereal box. You ca use a simulatio ivolvig rollig a six-sided die to determie the probability distributio for gettig a particular prize i four purchased boxes of cereal. 5. BINOMIAL DISTRIBUTIONS 9

3 Roll Roll Roll Roll Cout of s Defie a success as the appearace of a. What is the probability for oe, two, three, four, or zero s showig? The table i the margi shows the results of 0 repetitios of a simulatio of this experimet usig a spreadsheet. The frequecy ad experimetal probability distributio for all 00 repetitios of the simulatio appear below. Number of s Frequecy Experimetal Probability Aalysis of the Simulatio as a Coutig Problem Verify that the die-roll simulatio is a biomial experimet by checig that it has all the required properties. The, coduct the experimet yourself or do a simulatio to see if your experimetal probability distributio is the same.. There were four idetical trials i which a die was rolled. A trial ivolves rollig a die oe time. Each repetitio of the experimet cosists of four trials. You would the cout the umber of trials that produces a umber, i each of the 00 repetitios of the experimet.. The trials were Beroulli trials because there were oly two possible outcomes a success is a roll of, ad a failure is a roll that is ot a. I this case, p 6 ad p 5 6 ; the probability of success is the same for every roll; the trials are idepedet of oe aother; ad the purpose of the experimet is to determie the umber of s that occurs i four rolls. That is the biomial radom variable. Thi about how to record the possible outcomes of the four rolls of the die. For each repetitio of the experimet, record the outcome of each of the four trials i a table similar to the followig example Probability Distributio for Number of s 0 Number of s Roll Roll Roll Roll???? 9 CHAPTER 5 PROBABILITY DISTRIBUTIONS AND PREDICTIONS

4 Example Rollig s (a) What is the probability that the first roll will be a ad all the others will be somethig other tha a? (b) Fid the probability of the combied evet that the roll of will appear i ay of the four available positios i the table. (c) What is the probability that exactly two s show i the four rolls of the die? (d) Complete the theoretical probability distributio for the umber of s showig i four rolls. Solutio (a) Cosider the followig: P(Roll ) 6 P(Roll ) 5 6 Roll Roll Roll Roll??? P(Roll ) 5 6 P(Roll ) 5 6 Thus, P([Roll ] AND [Roll ] AND [Roll ] AND [Roll ]) Thi about? The Sum of the Probabilities i the Table Why do the probabilities add to? (b) The umber of ways that ca be placed i oe of the four etries of the table is the same as coutig the umber of ways oe object ca be selected from four available objects. This ca be doe i ways. Therefore, the probability that oly oe of the rolls will result i a showig is the sum of the four idividual probabilities, all of which are Therefore, P(oe i four trials) (c) Suppose the two s appeared i Roll ad Roll. The probability of this evet is or about 0.9 P([R ] AND [R ] AND [R ] AND [R ]) The umber of ways that exactly two of the etries ca 6. be filled i with s is Therefore, P(two s i four trials) or about 0.. Number of s Roll Roll Roll Roll?? 0 Probability BINOMIAL DISTRIBUTIONS 95

5 ? Thi about The Coefficiets What is the relatioship betwee these probabilities ad the Biomial Theorem? How does the expasio of (p + q) relate to the biomial probability distributio? (d) Each row of the probability distributio for this experimet ca be foud usig reasoig similar to that o the previous page. The probability that the four rolls of the die will show s is give by the formula P( s i four trials) This formula is aother way to express the probability distributio of the biomial radom variable that idicates the umber of s showig i four trials. Remember that a probability distributio ca be represeted by a graph, a table of values, or a formula. BINOMIAL PROBABILITY DISTRIBUTION I geeral, the probability of successes i trials of a biomial experimet correspods to fidig the umber of ways the successes ca be recorded i the available recordig slots i the outcome table for the evet. There are ways of selectig the locatios i which to record the successes. The probability of each of these is (p) (q) (p) ( p). Usig the Additive Priciple for Probabilities, the probability of the evet correspodig to successes is the sum of all the idividual probabilities for the outcomes that mae up the evet. Biomial Probability Distributio Cosider a biomial experimet i which there are Beroulli trials, each with a probability of success of p. The probability of successes i the trials is give by P(X ) (p) ( p) where X is the discrete radom variable correspodig to the umber of successes. Example Cost of Coffee I Sectio., you ivestigated the amout you could expect to pay for coffee if you ad a fried tossed a coi to determie who would pay each day. Coffee costs $.00 a cup. Determie the expected cost to you each wee. Solutio Assume that each coi toss is a Beroulli trial. Your correct call is defied as a success for which p. The experimet cosists of five trials (oe for each day of the wee). Cosider the umber of successes. The discrete radom variable, X, represets the umber of wis i five tosses. This is a biomial experimet for which the biomial probability distributio formula yields 5 P(X ) CHAPTER 5 PROBABILITY DISTRIBUTIONS AND PREDICTIONS

6 Probability E(X ) Probability Distributio for Coffee Game Coi Tosses 0 5 Number of Wis If we examie the expected cost of the coffee game, we also observe the followig: E(X ) 5.5 I this case, the umber of trials multiplied by the probability of success results i the expected value. 5 Expected Value of a Biomial Experimet The expected value of a biomial experimet that cosists of Beroulli trials with a probability of success, p, o each trial is E(X) p Example Probability of a Hit I Sectio., you cosidered a baseball player who had a battig average of 0.0. You created a simulatio to predict the lielihood that he could have a game i which he has o hits i three times at bat. (a) Determie the theoretical probability of this evet. (b) Determie the probability distributio for the umber of hits per game. (c) Calculate the batter s expected umber of hits per game. Solutio Assume that each time at bat is a Beroulli trial for which a success is a hit ad for which p 0.0. The experimet cosists of three trials i which the umber of hits is cosidered a success. For the calculatio of the theoretical probability, the umber of games played durig the seaso is ot relevat. 5. BINOMIAL DISTRIBUTIONS 97

7 Thi about? The Probability of No Hits Why is it reasoable for a batter with a 0.0 average to have a relatively small probability of gettig o hits i three times at bat? The discrete radom variable, X, represets the umber of hits i three times at bat. (a) This is a biomial experimet for which the biomial distributio formula yields P(X 0) (0.0)0 ( 0.0) 0. (b) The graph of the distributio is show below. 0 Probability Probability Distributio for Hits i Three At-Bats 0 Number of Hits? Thi about The Expected Value Why does a batter with a hittig average of 0.0 have a expected umber of hits i three times at bat that is almost? What would the batter s average be to mae the expected value exactly? If X is the umber of hits i three times at bat, the geeral formula for this distributio is (c) Method E(X) X i P(X i ) i P(X ) (0.0) (0.680) 0P(X 0) P(X ) P(X ) P(X ) 0.96 Method E(X) p (0.) 0.96 KEY IDEAS biomial experimet ay experimet that cosists of Beroulli trials ad for which the purpose of the experimet is to determie the umber of successes that occurs durig the trials 98 CHAPTER 5 PROBABILITY DISTRIBUTIONS AND PREDICTIONS

8 Beroulli trial Beroulli trials have the followig properties: There are two possible outcomes for each trial, which are usually termed success ad failure. The probability of a success is usually deoted p ad the probability of a failure is q or p. The probability of the outcomes remais the same from trial to trial. The values of p ad p do ot chage from oe trial to the ext. The trials are idepedet of oe aother. biomial probability distributio cosider a biomial experimet i which there are Beroulli trials, each with a probability of success of p. The probability of successes i the trials is give by P(X ) (p) ( p) where X is the discrete radom variable correspodig to the umber of successes expected value of a biomial experimet i ay biomial experimet, the expected umber of successes is give by E(X) x i P(X x i ) p i where is the umber of trials ad p is the probability of success o each trial 5. Exercises A B. For each term, idetify (i) the umber of trials (ii) the probability p of a success (iii) the umber of successes 0 (a) 6 7 (b). Evaluate the followig sum. 0. Suppose p. Simplify this expressio. (p) ( p). Explai why this sum is equal to. (p)0 ( p) 0 (p) ( p) (p) ( p) BINOMIAL DISTRIBUTIONS 99

9 5. Loo bac at Example i this sectio. Use it to aswer the followig questios. (a) The expressio for the probability of the specific evet that a appears oly i the first row of the outcome table has a probability (b) (c) (d) (e) (f) Explai. Why is the probability i part (a) multiplied by to fid the probability of gettig a sigle? Determie the probability that a eve umber shows i oly the secod ad third of four rolls of a six-sided die. Determie the probability for a eve umber showig i exactly two of the four rolls of a six-sided die. Determie the probability distributio for the umber of times a eve umber shows i four rolls of a six-sided die. Suppose you were tossig a coi four times istead of rollig a die. What is the probability of gettig a head o the first toss oly? 6. Commuicatio Explai why the problems i parts (a) ad (b) ca be modelled usig a biomial distributio, but the problems i parts (c) ad (d) caot. (a) Fid the probability that a customer will see a refud because of a defective product whe traditioally 0% of all customers have requested such a refud. (b) Fid the probability that defective parts will show up i a sample of 0 parts selected radomly from a maufacturig process that the plat ows has a 5% defect rate. (c) Fid the probability that defective parts will show up i a sample of 0 parts selected radomly from a maufacturig process whe it is ow that there are defective machies out of 0. (d) A hocey goalteder has stopped 87 of 00 shots. Fid the probability that she will stop the ext shots i a row. Desig simulatios for Questios 7 to. Compare the probability distributio resultig from your simulatio with the theoretical distributio i each questio. 7. Kowledge ad Uderstadig Mail-order maretig compaies have a respose rate of 5% to their advertisig flyers. (a) Compute the probability that exactly people out of a sample of 0 respod to the flyers they receive. (b) Fid the expected umber of people i a sample of 0 who will respod to the flyers. (c) Compute the probability that at least people out of a sample of 0 respod to the flyers they receive. 8. A family hopes to have six childre. Assume boys ad girls are bor with the same probability. (a) Determie the probability that four of the childre will be boys. (b) Determie the probability that at least two of the childre will be girls. (c) Determie the probability that all six childre will be girls. 00 CHAPTER 5 PROBABILITY DISTRIBUTIONS AND PREDICTIONS

10 9. Thiig, Iquiry, Problem Solvig A study published i a cosumer magazie idicated that whe a husbad ad a wife shop for a car, the husbad exerts the primary ifluece i the decisio 70% of the time. Five couples who will be purchasig a car are selected at radom. Determie the probability of each of the followig. (a) I exactly two of the couples, the husbad will exert the primary ifluece o the decisio. (b) I all five couples, the husbad will exert the primary ifluece o the decisio. (c) Fid the expected umber of couples i which the husbad will exert primary ifluece. (d) Determie the probability that i all five couples, the wife will exert the primary ifluece o the decisio. 0. A baseball player has a battig average of (a) Fid the probability that the player will get (i) at least hits i her ext 5 times at bat (ii) at least hits i her ext 0 times at bat (iii) at least 6 hits i her ext 0 times at bat (b) What is the player s expected umber of hits i her ext 0 times at bat?. I a large maufacturig plat, radom samples of 0 fial products are tae each hour. Whe a hourly defect rate exceedig out of 0 items is detected, productio is shut dow. (a) If a productio lot has a 0% defect rate, what is the probability that productio will be shut dow? (b) What is the probability that productio will be shut dow if the actual defect rate is 0%? (c) What is the expected umber of defective items i a sample of 0 for each defect rate i parts (a) ad (b)? Questios,, ad also appeared i Sectio.. For each questio, compare the simulatio results you obtaied i Sectio. with the theoretical probability that you fid here.. A field-goal icer for a high school football team has a 80% success rate based o his attempts this year. Determie the probability that he will miss three field goals i a row.. Te percet of the eyboards a computer compay maufactures are defective. Determie the probability that oe or more of the ext three eyboards to come off the assembly lie will be defective.. Applicatio Imagie that the first traffic light you ecouter o your way to school each morig has a 60-s cycle i which it is gree for 0 s. What is the probability that you will get a gree light o the ext three morig trips to school? 5. BINOMIAL DISTRIBUTIONS 0

11 ADDITIONAL ACHIEVEMENT CHART QUESTIONS 5. Kowledge ad Uderstadig Determie the probability, correct to four decimal places, that a die rolled six times i a row will produce the followig. (a) oe (b) five s (c) at least two s 6. Applicatio A multiple-choice quiz has 0 questios. Each questio has four possible aswers. Sam is certai that he ows the correct aswer for Questios, 5, ad 8. If he guesses o the other questios, determie the probability that he passes the quiz. 7. Thiig, Iquiry, Problem Solvig I the dice game Yahtzee, a player has three tries at rollig some or all of a set of five dice. Each player is tryig to achieve results such as three of a id, two pairs, full house, so o. A yahtzee occurs whe a player rolls five of a id. If Cheryl rolls a pair of s o the first toss, ad the rolls oly the o-s showig o the subsequet two tosses, fid the probability that she gets a yahtzee. 8. Commuicatio What coditios must be satisfied i order for a experimet to be cosidered a biomial experimet? Describe a situatio that meets these coditios. Chapter Problem Are Frog Populatios Decliig? CP. Suppose that the populatios of the three species are distributed as show i the followig table. You capture a frog, ote its species ad geder, ad the release it. This process is repeated util you have captured ad recorded 50 frogs. Percet Species Geder Ratio of Total Marsh Species Populatio Males Females Bullfrog 0% 60% 0% Sprig peeper 50% 50% 50% Mi frog 0% 50% 50% (a) Determie the probability that there will be at least five female bullfrogs i the sample. (b) Determie the probability that there will ot be ay mi frogs i the sample. (c) Suppose that there were 0 sprig peepers i the sample. Determie whether this is uusual eough to cause you to recosider your origial estimate of their proportio of the frog populatio. 0 CHAPTER 5 PROBABILITY DISTRIBUTIONS AND PREDICTIONS

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