Hypergeometric Distributions


 Rafe Fitzgerald
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1 7.4 Hypergeometric Distributios Whe choosig the startig lieup for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you deal cards from a stadard deck, there ca be o repetitios. I such situatios, each selectio reduces the umber of items that could be selected i the ext trial. Thus, the probabilities i these trials are depedet. Ofte we eed to calculate the probability of a specific umber of successes i a give umber of depedet trials. INVESTIGATE & INQUIRE: Choosig a Jury I Otario, a citize ca be called for jury duty every three years. Although most juries have 12 members, those for civil trials i Otario usually require oly 6 members. Suppose a civilcourt jury is beig selected from a pool of 18 citizes, 8 of whom are me. Develop a simulatio to determie the probability distributio for the umber of wome selected for this jury. 1. Select a radomumber geerator to simulate the selectio process. 2. Decide how to simplify the selectio process. Decide, also, whether the full situatio eeds to be simulated or whether a proportio of the trials would be sufficiet. 3. Desig each trial so that it simulates the actual situatio. Esure that each trial is depedet by settig the radomumber geerator so that there are o repetitios withi each series of trials. 4. Set up a method to record the umber of successes i each experimet. Pool your results with those of other studets i your class, if ecessary. 5. Use the results to estimate the probabilities of x successes (wome) i r trials (selectios of a juror). 6. Reflect o the results. Do they accurately represet the probability of x wome beig selected? 7. Compare your simulatio ad its results with those of your classmates. Which are the better simulatios? Explai why. Data i Actio The cost of ruig the crimial, civil, ad family courts i Otario was about $310 millio for These courts have the equivalet of 3300 fulltime employees. 7.4 Hypergeometric Distributios MHR 397
2 The simulatio i the ivestigatio models a hypergeometric distributio. Such distributios ivolve a series of depedet trials, each with success or failure as the oly possible outcomes. The probability of success chages as each trial is made. The radom variable is the umber of successful trials i a experimet. Calculatios of probabilities i a hypergeometric distributio geerally require formulas usig combiatios. Example 1 Jury Selectio a) Determie the probability distributio for the umber of wome o a civilcourt jury selected from a pool of 8 me ad 10 wome. b) What is the expected umber of wome o the jury? Solutio 1 Usig Pecil ad Paper a) The selectio process ivolves depedet evets sice each perso who is already chose for the jury caot be selected agai. The total umber of ways the 6 jurors ca be selected from the pool of 18 is (S ) = 18 C 6 = There ca be from 0 to 6 wome o the jury. The umber of ways i which x wome ca be selected is 10 C x. The me ca fill the remaiig 6 x positios o the jury i 8 C 6 x ways. Thus, the umber of ways of selectig a jury with x wome o it is 10 C x 8 C 6 x ad the probability of a jury with x wome is P(x) = (x) (S) = 10C C x 8 6 x P(x) 0.4 This combiatio could also be writte as C(18, 6) or Number of Wome, x Probability, P(x) 0 C C = C C = C C = Probability Number of Wome Jurors, x 3 C C = C C = C C = C C = MHR Probability Distributios
3 b) E(X ) = 6 x i P(x i ) i=0 = (0)( ) + (1)( ) + (2)( ) + (3)( ) + (4)( ) + (5)( ) + (6)( ) = The expected umber of wome o the jury is approximately Solutio 2 Usig a Graphig Calculator a) Eter the possible values for x, 0 to 6, i L1. The, eter the formula for P(x) i L2: (10 Cr L1) (8 Cr (6 L1)) (18 Cr 6) b) Calculate xp(x) i L3 usig the formula L1 L2. QUIT to the home scree. You ca fid the expected umber of wome by usig the sum( fuctio i the LIST MATH meu. The expected umber of wome o the jury is approximately Solutio 3 Usig a Spreadsheet a) Ope a ew spreadsheet. Create titles x, p(x), ad xp(x) i colums A to C. Eter the values of the radom variable x i colum A, ragig from 0 to 6. Next, use the combiatios fuctio to eter the formula for P(x) i cell B3 ad copy it to cells B4 through B Hypergeometric Distributios MHR 399
4 b) Calculate xp(x) i colum C by eterig the formula A3*B3 i cell C3 ad copyig it to cells C4 through C9. The, calculate the expected value usig the SUM fuctio. The expected umber of wome o the jury is approximately Solutio 4 Usig Fathom TM Ope a ew Fathom documet. Drag a ew collectio box to the work area ad ame it Number of Wome Jurors. Create seve ew cases. Drag a ew case table to the work area. Create three ew attributes: x, px, ad xpx. Eter the values from 0 to 6 for the x attribute. Rightclick o the px attribute, select Edit Formula, ad eter combiatios(10,x)*combiatios(8,6x)/combiatios(18,6) Similarly, calculate xp(x) usig the formula x*px. Next, doubleclick o the collectio box to ope the ispector. Select the Measures tab, ad ame a ew measure Ex. Rightclick o Ex ad use the sum fuctio to eter the formula sum(x*px). The expected umber of wome o the jury is approximately MHR Probability Distributios
5 You ca geeralize the methods i Example 1 to show that for a hypergeometric distributio, the probability of x successes i r depedet trials is Probability i a Hypergeometric Distributio P(x) = ac C x a r x C, r where a is the umber of successful outcomes amog a total of possible outcomes. Although the trials are depedet, you would expect the average probability of a a success to be the same as the ratio of successes i the populatio,. Thus, the expectatio for r trials would be Expectatio for a Hypergeometric Distributio E(X ) = r a This formula ca be prove more rigorously by some challegig algebraic maipulatio of the terms whe P(x) = C C a x a r x is substituted ito the equatio for the expectatio of ay probability distributio, E(X ) = x i P(x i ). C r i=1 Example 2 Applyig the Expectatio Formula Calculate the expected umber of wome o the jury i Example 1. Solutio E(X ) = r a = = 3.33 The expected umber of wome jurors is Example 3 Expectatio of a Hypergeometric Distributio A box cotais seve yellow, three gree, five purple, ad six red cadies jumbled together. a) What is the expected umber of red cadies amog five cadies poured from the box? b) Verify that the expectatio formula for a hypergeometric distributio gives the same result as the geeral equatio for the expectatio of ay probability distributio. 7.4 Hypergeometric Distributios MHR 401
6 Solutio a) = r = 5 a = 6 = 21 Usig the expectatio formula for the hypergeometric distributio, E(X ) = r a 5 6 = 21 = Oe would expect to have approximately 1.4 red cadies amog the 5 cadies. b) Usig the geeral formula for expectatio, E(X ) = xp(x) = (0) C C + (1) C C 6 + (2) C C + (3) C C + (4) C C + (5) C C = Agai, the expected umber of red cadies is approximately 1.4. Example 4 Wildlife Maagemet I the sprig, the Miistry of the Eviromet caught ad tagged 500 raccoos i a wilderess area. The raccoos were released after beig vacciated agaist rabies. To estimate the raccoo populatio i the area, the miistry caught 40 raccoos durig the summer. Of these 15 had tags. a) Determie whether this situatio ca be modelled with a hypergeometric distributio. b) Estimate the raccoo populatio i the wilderess area. To lear more about samplig ad wildlife, visit Solutio the above web site ad follow the liks. Write a brief descriptio of some of the samplig a) The 40 raccoos captured durig the summer were all differet from each other. I other words, techiques that are used. there were o repetitios, so the trials were depedet. The raccoos were either tagged (a success) or ot (a failure). Thus, the situatio does have all the characteristics of a hypergeometric distributio. b) Assume that the umber of tagged raccoos caught durig the summer is equal to the expectatio for the hypergeometric distributio. You ca substitute the kow values i the expectatio formula ad the solve for the populatio size,. 402 MHR Probability Distributios
7 Here, the umber of raccoos caught durig the summer is the umber of trials, so r = 40. The umber of tagged raccoos is the umber of successes i the populatio, so a = 500. ra E(X ) =, so 15= = = The raccoo populatio i the wilderess area is approximately Alteratively, you could assume that the proportio of tagged raccoos amog the sample captured durig the summer correspods to that i the whole populatio. The, 1 5 = 50 0, which gives the same estimate for 40 as the calculatio show above. Key Cocepts A hypergeometric distributio has a specified umber of depedet trials havig two possible outcomes, success or failure. The radom variable is the umber of successful outcomes i the specified umber of trials. The idividual outcomes caot be repeated withi these trials. The probability of x successes i r depedet trials is P(x) = C C a x a r x, where is the populatio size ad a is the umber of successes i the populatio. ra The expectatio for a hypergeometric distributio is E(X ) =. To simulate a hypergeometric experimet, esure that the umber of trials is represetative of the situatio ad that each trial is depedet (o replacemet or resettig betwee trials). Record the umber of successes ad summarize the results by calculatig probabilities ad expectatio. C r Commuicate Your Uderstadig 1. Describe how the graph i Example 1 differs from the graphs of the uiform, biomial, ad geometric distributios. 2. Cosider this questio: What is the probability that 5 people out of a group of 20 are left haded if 10% of the populatio is lefthaded? Explai why this situatio does ot fit a hypergeometric model. Rewrite the questio so that you ca use a hypergeometric distributio. 7.4 Hypergeometric Distributios MHR 403
8 Practise A 1. Which of these radom variables have a hypergeometric distributio? Explai why. a) the umber of clubs dealt from a deck b) the umber of attempts before rollig a six with a die c) the umber of 3s produced by a radomumber geerator d) the umber of defective screws i a radom sample of 20 take from a productio lie that has a 2% defect rate e) the umber of male ames o a page selected at radom from a telephoe book f) the umber of lefthaded people i a group selected from the geeral populatio g) the umber of lefthaded people selected from a group comprised equally of lefthaded ad righthaded people 2. Prepare a table ad a graph of a hypergeometric distributio with a) = 6, r = 3, a = 3 b) = 8, r = 3, a = 5 Apply, Solve, Commuicate B 3. There are five cats ad seve dogs i a pet shop. Four pets are chose at radom for a visit to a childre s hospital. a) What is the probability that exactly two of the pets will be dogs? b) What is the expected umber of dogs chose? 4. Commuicatio Earlier this year, 520 seals were caught ad tagged. O a recet survey, 30 out of 125 seals had bee tagged. a) Estimate the size of the seal populatio. b) Explai why you caot calculate the exact size of the seal populatio. 5. Of the 60 grade12 studets at a school, 45 are takig Eglish. Suppose that 8 grade12 studets are selected at radom for a survey. a) Develop a simulatio to determie the probability that 5 of the selected studets are studyig Eglish. b) Use the formulas developed i this sectio to verify your simulatio results. 6. Iquiry/Problem Solvig I a study of Caada geese, 200 of a kow populatio of 1200 geese were caught ad tagged. Later, aother 50 geese were caught. a) Develop a simulatio to determie the expected umber of tagged geese i the secod sample. b) Use the formulas developed i this sectio to verify your simulatio results. 7. Applicatio I a mathematics class of 20 studets, 5 are biligual. If the class is radomly divided ito 4 project teams, a) what is the probability that a team has fewer tha 2 biligual studets? b) what is the expected umber of biligual studets o a team? 8. I a swim meet, there are 16 competitors, 5 of whom are from the Easter Swim Club. a) What is the probability that 2 of the 5 swimmers i the first heat are from the Easter Swim Club? b) What is the expected umber of Easter Swim Club members i the first heat? 9. The door prizes at a dace are four $10 gift certificates, five $20 gift certificates, ad three $50 gift certificates. The prize evelopes are mixed together i a bag, ad five prizes are draw at radom. a) What is the probability that oe of the prizes is a $10 gift certificate? b) What is the expected umber of $20 gift certificates draw? 404 MHR Probability Distributios
9 10. A 12member jury for a crimial case will be selected from a pool of 14 me ad 11 wome. a) What is the probability that the jury will have 6 me ad 6 wome? b) What is the probability that at least 3 jurors will be wome? c) What is the expected umber of wome? 11. Seve cards are dealt from a stadard deck. a) What is the probability that three of the seve cards are hearts? b) What is the expected umber of hearts? 12. A bag cotais two red, five black, ad four gree marbles. Four marbles are selected at radom, without replacemet. Calculate a) the probability that all four are black b) the probability that exactly two are gree c) the probability that exactly two are gree ad oe are red d) the expected umbers of red, black, ad gree marbles Kowledge/ Uderstadig ACHIEVEMENT CHECK Thikig/Iquiry/ Problem Solvig Commuicatio Applicatio 13. A calculator maufacturer checks for defective products by testig 3 calculators out of every lot of 12. If a defective calculator is foud, the lot is rejected. a) Suppose 2 calculators i a lot are defective. Outlie two ways of calculatig the probability that the lot will be rejected. Calculate this probability. b) The qualitycotrol departmet wats to have at least a 30% chace of rejectig lots that cotai oly oe defective calculator. Is testig 3 calculators i a lot of 12 sufficiet? If ot, how would you suggest they alter their qualitycotrol techiques to achieve this stadard? Support your aswer with mathematical calculatios. C 14. Suppose you buy a lottery ticket for which you choose six differet umbers betwee 1 ad 40 iclusive. The order of the first five umbers is ot importat. The sixth umber is a bous umber. To wi first prize, all five regular umbers ad the bous umber must match, respectively, the radomly geerated wiig umbers for the lottery. For the secod prize, you must match the bous umber plus four of the regular umbers. a) What is the probability of wiig first prize? b) What is the probability of wiig secod prize? c) What is the probability of ot wiig a prize if your first three regular umbers match wiig umbers? 15. Iquiry/Problem Solvig Uder what coditios would a biomial distributio be a good approximatio for a hypergeometric distributio? 16. Iquiry/Problem Solvig You start at a corer five blocks south ad five blocks west of your fried. You walk orth ad east while your fried walks south ad west at the same speed. What is the probability that the two of you will meet o your travels? 17. A research compay has 50 employees, 20 of whom are over 40 years old. Of the 22 scietists o the staff, 12 are over 40. Compare the expected umbers of older ad youger scietists i a radomly selected focus group of 10 employees. 7.4 Hypergeometric Distributios MHR 405
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