1STAT 702/J702 B.Habing Univ. of S.C. 1STAT 702/J702 September 2nd, 2004Instructor: Brian HabingDepartment of StatisticsLeConte 203Telephone: 803-777-3578E-mail: [email protected] 702/J702 B.Habing Univ. of S.C. 2Today• Homework Solutions• The Hypergeometric• Binomial vs. HypergeometricSTAT 702/J702 B.Habing Univ. of S.C. 3Ch.1 # 38) A child has six blocks, three of which are red and three of which are green. How many patterns can she make by placing them in a line?What if three white blocks are added?2STAT 702/J702 B.Habing Univ. of S.C. 4Ch. 1 # 42) How many ways can 11 boys on a soccer team be grouped into 4 forwards, 3 midfielders, 3 defenders, and 1 goalie?STAT 702/J702 B.Habing Univ. of S.C. 5Ch. 1 # 57) Cabinets A, B, and C each have two drawers with one coin per drawer. A has two gold, B has two silver, and C has one gold and one silver. A cabinet is chosen at random and a drawer is opened showing a silver. What is the chance the other is silver too?STAT 702/J702 B.Habing Univ. of S.C. 6Last time… Binomial Experiment1.nidentical trials2. Each trial has only two possible outcomes (“Success” or “Failure”)3. Probability of “Success” is a constant pfor every trial4. Trials are independentknkppknnkP−−⎟⎟⎠⎞⎜⎜⎝⎛= )1(] trialsin successes [3STAT 702/J702 B.Habing Univ. of S.C. 7Hypergeometric Experiment1. Population of size n2. rare “successes” and n-rare “failures”3. A random sample of size mis taken without replacementSTAT 702/J702 B.Habing Univ. of S.C. 8Example) An assembly line produced n =2000 parts, of which r =40 were defective. (Note that this is a 0.02 defective rate).A random sample of size m =20 is chosen. What is the probability that exactly 10 of these 20 will be defectives?STAT 702/J702 B.Habing Univ. of S.C. 9The first “trick” is to realize that, since we are taking a random sample, every possible sample of size 20 has the same probability. (e.g. all of the sample points have the same probability.)In the binomial case we figured out the probability of each sample point and then multiplied that by the number of sample points in our event.4STAT 702/J702 B.Habing Univ. of S.C. 10Another way of calculating the probability of an event when all sample points are equally probable is:P(A) = number of sample points in Atotal number of sample pointsSTAT 702/J702 B.Habing Univ. of S.C. 11In general, for a population of size nwith ksuccesses and a sample of size mwe get:⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛−−⎟⎟⎠⎞⎜⎜⎝⎛=mnkmrnkrmkP ] ofout successes [STAT 702/J702 B.Habing Univ. of S.C. 12Example – Capture/Recapture)Goal: To estimate the size nof a population.Method: “Randomly” capture, tag, and release rof them. Then “randomly capture” mof them and see how many are tagged.5STAT 702/J702 B.Habing Univ. of S.C. 13Now the probability of a certain number being captured will be hypergeometric!⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛−−⎟⎟⎠⎞⎜⎜⎝⎛=mnkmrnkrmkP ] ofout tagged[STAT 702/J702 B.Habing Univ. of S.C. 14The problem is that we know r, k, and m, but we are looking for n!Since we can’t find nexactly, we will attempt to estimate it by choosing the value of nthat “seems most likely”. That is, what value of nwould give us the largest probability of observing the kthat we did.STAT 702/J702 B.Habing Univ. of S.C. 15Mathematically then, we need to find the nthat maximizesIf nwas continuous we could try taking the derivative with respect to nand setting it equal to zero.⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛−−⎟⎟⎠⎞⎜⎜⎝⎛==mnkmrnkrmkPLn] ofout tagged[6STAT 702/J702 B.Habing Univ. of S.C. 16We will use a similar logic here and take the ratio Ln/Ln-1.So nshould be the greatest integer not exceeding mr / k.So if r =10, m =20, and k =4 we estimate nto be 50. STAT 702/J702 B.Habing Univ. of S.C. 17When are the binomial and hypergeometric similar?STAT 702/J702 B.Habing Univ. of S.C. 18What do you lose if you sample with replacement instead? (e.g. why not always use