STAT 400 1 2 1 2 Discussion 4 Spring 2015 Suppose that the probability that a duck hunter will successfully hit a duck is 0 40 on any given shot Suppose also that the outcome of each shot is independent from the others a What is the probability that the first successful hit would be on the fourth shot b What is the probability that it would take at least six shots to hit a duck c What is the probability that the first successful hit would happen during an even numbered shot d What is the probability that the third successful hit would be on the ninth shot e What is the probability that the hunter would have three successful hits in nine shots f What is the probability that the hunter would have at least six successful hits in nine shots 3 Suppose that number of accidents at the Monstropolis power plant follows the Poisson process with the average rate of 0 40 accidents per day Assume all days are independent of each other a Find the probability that at least 2 accidents will occur in one day b Find the probability that there will be exactly 4 accidents in one week 7 days c Find the probability that there will be exactly 5 accident free days in one week 7 days 4 The birthday problem A probability class has N students a What is the probability that at least 2 students in the class have the same birthday For simplicity assume that there are always 365 days in a year and that birth rates are constant throughout the year b Use a computer or a calculator to find the smallest class size for which the probability that at least 2 students in the class have the same birthday exceeds 0 5 5 According to news reports in early 1995 among the first Pentium chips Intel made some had a peculiar defect which rendered some rarely carried out arithmetic operations incorrect Any chip could therefore be classified into one of three categories Good Broken useless or Defective operable except for the peculiar defect described above Suppose that 70 of the chips made were good 25 had a peculiar defect and 5 were broken If a random sample of 20 chips was selected what is the probability that 15 were good 3 defective and 2 broken 1 2 1 Suppose that the probability that a duck hunter will successfully hit a duck is 0 40 on any given shot Suppose also that the outcome of each shot is independent from the others a What is the probability that the first successful hit would be on the fourth shot Miss Miss Miss Hit 0 60 0 60 0 60 0 40 0 0864 Geometric distribution p 0 40 b P X 4 1 0 40 4 1 0 40 0 0864 What is the probability that it would take at least six shots to hit a duck For Geometric p P X a 1 p a a 0 1 2 P X 6 P X 5 0 60 5 0 07776 OR P X 6 1 P X 1 P X 2 P X 3 P X 4 P X 5 1 0 60 0 0 40 0 60 1 0 40 0 60 2 0 40 0 60 3 0 40 0 60 4 0 40 1 0 40 0 24 0 144 0 0864 0 05184 1 0 92224 0 07776 c What is the probability that the first successful hit would happen during an even numbered shot P even P 2 P 4 P 6 0 40 0 60 1 0 40 0 60 3 0 40 0 60 5 1 24 3 2k 0 24 0 60 0 24 0 36 n 0 24 1 0 36 64 8 0 375 k 0 n 0 OR P odd 0 40 0 60 1 0 40 0 60 3 0 40 0 60 5 P even 0 40 0 60 0 0 40 0 60 2 0 40 0 60 4 2 5 P even 3 P even 0 60 P odd 1 P odd P even d What is the probability that the third successful hit would be on the ninth shot P odd 8 P even 3 P even 3 0 375 8 Negative Binomial distribution p 0 40 r 3 8 shots 2 S s 6 F s S 8 0 40 2 0 60 6 0 40 0 0836 2 OR SSFFFFFFS SFSFFFFFS SFFSFFFFS SFFFSFFFS SFFFFSFFS SFFFFFSFS SFFFFFFSS FSSFFFFFS FSFSFFFFS FSFFSFFFS FSFFFSFFS FSFFFFSFS FSFFFFFSS FFSSFFFFS 28 0 40 3 0 60 6 0 0836 FFSFSFFFS FFSFFSFFS FFSFFFSFS FFSFFFFSS FFFSSFFFS FFFSFSFFS FFFSFFSFS FFFSFFFSS FFFFSSFFS FFFFSFSFS FFFFSFFSS FFFFFSSFS FFFFFSFSS FFFFFFSSS e What is the probability that the hunter would have three successful hits in nine shots Let X the number of successful hits in 9 shots Then X has Binomial distribution n 9 p 0 40 n P X k p k 1 p n k k Need P X 3 9 P X 3 0 40 3 0 60 6 0 2508 3 OR SSFFFFFFS SFSFFFFFS SFFSFFFFS SFFFSFFFS SFFFFSFFS SFFFFFSFS SFFFFFFSS FSSFFFFFS FSFSFFFFS FSFFSFFFS FSFFFSFFS FSFFFFSFS FSFFFFFSS FFSSFFFFS FFSFSFFFS FFSFFSFFS FFSFFFSFS FFSFFFFSS FFFSSFFFS FFFSFSFFS FFFSFFSFS FFFSFFFSS FFFFSSFFS FFFFSFSFS FFFFSFFSS FFFFFSSFS FFFFFSFSS FFFFFFSSS SSSFFFFFF SSFSFFFFF SSFFSFFFF SSFFFSFFF SSFFFFSFF SSFFFFFSF SFSSFFFFF SFSFSFFFF SFSFFSFFF SFSFFFSFF SFSFFFFSF SFFSSFFFF SFFSFSFFF SFFSFFSFF SFFSFFFSF SFFFSSFFF SFFFSFSFF SFFFSFFSF SFFFFSSFF SFFFFSFSF SFFFFFSSF FSSSFFFFF FSSFSFFFF FSSFFSFFF FSSFFFSFF FSSFFFFSF FSFSSFFFF FSFSFSFFF FSFSFFSFF FSFSFFFSF FSFFSSFFF FSFFSFSFF FSFFSFFSF FSFFFSSFF FSFFFSFSF FSFFFFSSF FFSSSFFFF FFSSFSFFF FFSSFFSFF FFSSFFFSF FFSFSSFFF FFSFSFSFF FFSFSFFSF FFSFFSSFF FFSFFSFSF FFSFFFSSF FFFSSSFFF FFFSSFSFF FFFSSFFSF FFFSFSSFF FFFSFSFSF FFFSFFSSF FFFFSSSFF FFFFSSFSF FFFFSFSSF FFFFFSSSF 84 0 40 3 0 60 6 0 2508 OR P X 3 P X 3 P X 2 CDF 3 CDF 2 0 483 0 232 0 251 f What is the probability that the hunter would have at least six successful hits in nine shots P X 6 1 P X 5 1 CDF 5 1 0 901 0 099 OR 9 9 P X 6 0 40 6 0 60 3 0 40 7 0 60 2 6 7 9 9 0 40 8 0 60 1 0 40 9 0 60 0 8 9 0 07432 0 02123 0 00354 0 00026 0 09935 3 Suppose that number of accidents at the Monstropolis power plant follows the Poisson process with the average rate of 0 40 accidents per day Assume all days are independent of each other a Find the probability that at least 2 accidents will occur in one day 1 day 0 40 Need P X 2 P X x Poisson distribution x e x 0 40 0 e 0 40 0 40 1 e 0 40 P X 2 1 P X 0 P X 1 1 0 1 1 0 67032 0 26813 1 0 93845 0 06155 b Find the probability that there will be exactly 4 accidents in one week 7 days 7 days 0 40 7 2 8 c P X 4 2 8 4 e 2 8 0 15574 4 Find the probability that there will be exactly 5 accident free days in one …
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