Unformatted text preview: Ch. 3: Combinatorial Probability High h2p://brownsharpie.courtneygibbons.org/?cat=22 Sampling With Replacement (BCR): Example Suppose that a sample of size 2 is drawn with replacement from a populaIon of size 5. a) Use a direct lisIng to determine the number of possible ordered pairs. b) Solve part (a) by using BCR. aa ab ac ad ae ba bb bc bd be ca cb cc cd ce da db dc dd de ea eb ec ed ee Sampling Without Replacement (PermutaIon): Example Suppose that a sample of size 2 is drawn without replacement from a populaIon of size 5. a) Use a direct lisIng to determine the number of possible ordered pairs. b) Solve part (a) by using BCR. aa ab ac ad ae ba bb bc bd be ca cb cc cd ce da db dc dd de ea eb ec ed ee Sampling Without Replacement (CombinaIon): Example Suppose that a sample of size 2 is drawn without replacement from a populaIon of size 5. Use a direct lisIng to determine the number of possible unordered pairs. aa ab ac ad ae ba bb bc bd be ca cb cc cd ce da db dc dd de ea eb ec ed ee CombinaIons Rule: Example In an a2empt to a2ract people to buying Kindle books, a merchandiser states that if a person buys 4 Kindle books, that person will get two free. Currently, this merchandiser has 60 Kindle books in stock. How many possibiliIes does the person have to selecIng the 6 books? a) How many diﬀerent poker hands do you have in 5 card draw? b) How many diﬀerent hands are there with 2 kings and 2 queens and one other card? c) How many diﬀerent hands are there with 2 pair? Ordered ParIIon: Example a) List all of the possible ordered parIIons of these 5 le2ers into two disInct groups of sizes 3 and 2. {abc},{de} {abd},{ce} {abe},{cd} {acd},{be} {ace},{bd} {ade},{bc} {bcd},{ae} {bce},{ad} {bde},{ac} {cde},{ab} b) Use part (a) to determine the number of possible ordered parIIons of the 5 le2ers into the two groups. c) Use the combinaIons rule and BCR to determine the number of possible ordered parIIons of the 5 le2ers into the 2 groups. Ordered ParIIon: Example 2 a) If you want to parIIon your class of 30 students into 7 groups, how many possible ways can you do this if the group sizes are 3, 4, 5, 5, 5, 4, and 4? b) How many diﬀerent possible hands are there in a 5 card draw poker game with 5 players? CounIng Rules and ProbabiliIes: Example 3.18 A drug known to have a 50% eﬀecIveness rate in curing a disease is administered to 20 people who have the disease. Determine the probability that a) exactly 10 of the 20 people will be cured. b) at least three fourths of the 20 people will be cured. CounIng Rules and ProbabiliIes: 1. In tossing 5 6 sided fair dice, what is the probability of at least one 2? 2. What is the probability that at least two students in this class, size = 26, have the same birthday? Coincidences …Once we set aside coincidences having apparent causes, four principles account for large numbers of remaining coincidences: hidden cause; psychology, including memory and percepIon; mulIplicity of endpoints, including the counIng of "close" or nearly alike events as if they were idenIcal; and the law of truly large numbers, which says that when enormous numbers of events and people and their interacIons cumulate over Ime, almost any outrageous event is bound to occur. These sources account for much of the force of synchronicity. (Abstract) ….. The probability problems discussed in SecIon 7 make the point that in many problems our intuiIve grasp of the odds is far oﬀ. We are ofen surprised by things that turn out to be fairly likely occurrences. (IntroducIon) Diaconis, P. and Mosteller, F. "Methods for Studying Coincidences." J. Amer. Sta+st. Assoc. 84, 853 861, 1989 CounIng Rules and ProbabiliIes: 1. In tossing 5 6 sided fair dice, what is the probability of at least one 2? 2. What is the probability that at least two students in this class, size = 26, have the same birthday? 3. What is the probability of a 2 pair in 5 card draw? ProbabiliIes – Random Sampling Example 3.20 For a random sample of size n with populaIon of size N, determine the probability that a speciﬁed member will be included in the sample in the case of an a) ordered sample with replacement b) ordered sample without replacement c) unordered sample without replacement Matching Problem: Example At a parIcular party there are N people, they all throw their coats into a pile. If each of the people select one coat from the pile at random, ﬁnd the probability that a) a speciﬁed person gets their own coat. b) n speciﬁed people get their own coats. c) at least one of the people get their own coat. Exercise 3.59 An ordinary deck of 52 playing cards is shuﬄed and dealt. What is the probability that a) the 7th card is an ace? b) the 7th card is the ﬁrst ace? What method to use? 1. How many banng orders are there for 9 players on a baseball team? [362,880] 2. In a math club at Purdue with 20 members, 3 people can go to a naIonal conference. How many diﬀerent ways can these people be chosen? [1140] 3. At a movie fesIval, a team of judges is to pick the ﬁrst, second, and third place winners from the 18 ﬁles entered. How many possible ways are there to choose the winners? [4896] 4. The Internal Revenue Service (IRS) decides that it will audit the returns of 3 from a group of 18. How many possible ways are there to choose the returns who will be audited? [816] What method to use? (cont) 5. The English alphabet has 26 le2ers. How many 4 le2er combinaIons can be formed? [456,976] 6. A staIsIcs professors wants to do perform in depth interviews to see how she is teaching. So she has decided to choose 5 students from her class of 40. How many diﬀerent possibiliIes are there? [658,008] 7. The menu at a restaurant has ﬁve choices of a beverage, three diﬀerent salads, sixe entrées and four deserts. How many diﬀerent meals are possible? [360] 8. The sales manager of a clothing company needs to assign seven salespeople to seven diﬀerent territories. How many possibiliIes are there for the assignments? [5040] ...
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