# Exercises 1. Determine whether or not the random variable X is a binomial random variable. If so, give the values of n and p. If not, explain why not. a. X is the number of dots on the top face of fair die that is rolled. b. X is the number of hearts in a five-card hand drawn (without replacement) from a well-shuffled ordinary deck. C. X is the number of defective parts in a sample of ten randomly selected parts coming from a manufacturing process in which 0.02% of all parts are defective. d. X is the number of times the number of dots on the top face of a fair die is even in six rolls of the die. e. X is the number of dice that show an even number of dots on the top face when six dice are rolled at once. Solution a. not binomial; not success/failure. b. not binomial; trials are not independent. C. binomial; n = 10, p = 0.0002 d. binomial; n = 6, p = 0.5 e. binomial; n = 6, p = 0.5 11 Scarred with ComScorer a = 2. X is a binomial random variable with parameters n= 12 and p = 0.82. Compute the probability indicated. a. P(11) b. P(9) C. PO) d. P(13) Solution a. 0.2434 b. 0.2151 C. 0.1812=0 d. O 17 Scarred with CamScorer a. 3. Xis a binomial random variable with the parameters shown. Use the special formulas to compute its mean p and standard deviation o. n=8, p= 0.43 b. n= 47, p= 0.82 1200, p= 0.44 d. n=2100, p= 0.62 2 C. n= Solution a. l = 3.44, o = 1.4003 A = 38.54, 0 = 2.6339 C. H = 528, 0 = 17.1953 d. H = 1302, J = 22.2432 13 Scarred with ComScorer 4. The probability that a certain kind of component will survive a shock test is 3/4. Find the probability that exactly 2 of the next 4 components tested survive. Solution Assuming that the tests are independent and p =3/4 for each of the 4 p tests, we obtain 2 2 32 27 P P(2) = ( 1) (*) * (1) *= (219) (*) – 12 4! ! ! = = 14 Scarred with CamScorer 5. A large chain retailer purchases a certain kind of electronic device from a manufacturer. The manufacturer indicates that the defective rate of the device is 3%. The inspector randomly picks 20 items from a shipment. What is the probability that there will be at least one defective item among these 20? Solution Denote by X the number of defective devices among the 20. P(X > 1) = 1 – P(X = 0) = = 1 – (0.03)º(1 – 0.03)20-0 = 0.4562. Scarred with CamScomer

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