STAT 400 Discussion 4 Spring 2015 1 – 2. 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. 1. 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? 2. 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. 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. 1. 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. P ( X = 4 ) = ( 1 – 0.40 ) 4 – 1 0.40 = 0.0864. b) 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) + … = +++⋅⋅⋅531 60.00.40 60.00.40 60.040.0… = 83==−=∞=∞⋅⋅⋅∑∑==642436.01124.036.024.060.024.0002 nnkk = 0.375. OR P(odd) = +++ ⋅⋅⋅531 60.00.40 60.00.40 60.040.0… P(even) = +++ ⋅⋅⋅420 60.00.40 60.00.40 60.040.0… ⇒ P(even) = 0.60 ⋅ P(odd). P(odd) = 35 ⋅ P(even). ⇒ 1 = P(odd) + P(even) = 38 ⋅ P(even). P(even) = 83 = 0.375. 2. d) What is the probability that the third successful hit would be on the ninth shot? Negative Binomial distribution, p = 0.40, r = 3. [ 8 shots: 2 S’s & 6 F’s ] S ( ) ( )⋅⋅ 60040028 62 .. ⋅ 0.40 ≈ 0.0836. OR S S F F F F F F S F S S F F F F F S F F S F S F F F S F F F S F F F S S S F S F F F F F S F S F S F F F F S F F S F F S F F S F F F F S S F F S S F F S F F F F S F S F F S F F F S F F S F F F S F S F F F F S F S F S S F F F S F F F S F S F F F S F F S F F S F F F F S S F F F F S F F S S S F F F F S F F S F S F F F F S F S F F F S S F F F S F F F F F S S F S S F F F F F S F S F S F F F F F S S F F F S F S F F S F F F F F S F S S S F F F F F F S S F F S S F F F F S F F F S F F S F S F F F F F F S S S 28 ⋅ (0.40) 3 ⋅ (0.60) 6 ≈ 0.0836.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. Need P( X = 3 ) = ? . 1 P)()( knkppknkX−−== P( X = 3 ) = 63)60.0()40.0(39 = 0.2508. OR S S F F F F F F S F S S F F F F F S F F S F S F F F S F F F S F F F S S S F S F F F F F S F S F S F F F F S F F S F F S F F S F F F F S S F F S S F F S F F F F S F S F F S F F F S F F S F F F S F S F F F F S F S F S S F F F S F F F S F S F F F S F F S F F S F …
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