s: Chapter 4: 24. a) A lot of 25 items is to be inspected by means of a two-stage sampling plan. A sample of five items is drawn. If one or more is bad, the lot is rejected. If all are non-defective, a second sample of 10 items is drawn from the remaining 20 items. The lot is then rejected if any item in the second sample is bad; otherwise it is accepted. Find the probability of accepting a lot which has two defective items. b) An alternative single-stage plan for this problem might be to choose a single sample of 15 items and accept the lot only if there are no defectives in the sample. Find the probability of accepting a lot of 25 with two defectives and compare this probability with that found in a). 30. A box contains three defective and seven non-defective light bulbs. a) If three bulbs are drawn consecutively at random (WOR), develop a three-stage model (outcomes and probabilities). You may find a tree diagram useful. b) Determine the probability that i) At least one non-defective bulb is drawn. ii) All three are non-defective if the first is non-defective. 34. A fruit store obtains its supply of apples from three producers X, Y and Z. Producer X supplies 20% of the requirements and on the average 2% of his apples are bruised. Producers Y and Z supple 30% and 50%, respectively. On the average 1% of Y's and 5% of Z's apples are bruised. In a given week the fruit store owner purchases 5000 apples. If an apple chosen at random turns out to be bruised, what is the probability that it was supplied by Y? 36. Two closely related species of mushrooms (I and II) are difficult to identify without the aid of a microscope. One method used in the field to separate the two species is to note the presence or absence of a ring on the stalk of the plant. Ninety percent of species I and 20% of species II have the ring. It is also known that in the particular area where the mushrooms are being studied, 70% of them are species I. a) Suppose the field worker finds a mushroom with a ring and decides it belongs to species I. What is the probability that he is correct? b) If mushrooms with rings are classified as species I and those without as species II, what proportion of mushrooms will be correctly classified? Chapter 6: 1. A coin is weighted so that P(H) = 3/8 and P(T) = 5/8. The coin is tossed until a head or three tails appear. a) Describe a suitable sample space for this experiment and assign appropriate probabilities to each outcome. What assumptions have you made? b) Let X be the number of tosses needed. Find the p.f. and d.f. of X. c) Graph both the p.f. and d.f. obtained in part b). 4. A tetrahedron is tossed into the air and the bottom face on which it comes to rest is noted. Each of the four faces-numbered 1,2,3,4 has an equal probability of being on the bottom when the tetrahedron comes to rest. Suppose the tetrahedron is tossed twice. a) What is an appropriate sample space for this experiment? What is the probability associated with each outcome? Discuss carefully any consumptions which you have made in assigning these probabilities. b) Define a random variable X to be the sum of the outcomes on the two tosses. Find the p.f. of X. 16. Two dead batteries have accidentally been placed in a box with six good batteries. Four batteries are drawn (WOR) from the box. a) Determine the p.f. of X, the number of dead batteries selected b) Find the d.f. of X and graph it. Chapter 7 23. Suppose one dead battery has been put into a box with three good ones. If the batteries are tested one at a time until the dead one is found, find the expected number of batteries which have been tested when the defective is found. 24. A lottery has a first prize of $100, a second prize of $50 and four $25 prizes. Would you pay $1 for a ticket if there were a) 100 tickets being sold b) 500 tickets being sold 25. A game played by two people is said to be fair if the expected return for each player is zero. a) Suppose Tom and Jerry roll a balanced die and Tom agrees to pay Jerry $5 if the score is less than 3. How much should Jerry pay Tom when the score is greater than or equal to 3 if the game is to fair? b) If the game in a) is altered so that no one wins any money if a 3 appears, what should Jerry pay Tom so that the game will be fair? 26. Jack pays Bill $1 and two fair dice are rolled. Jack receives $2 from Bill if one 6 appears, $4 if two 6's appears and no return otherwise. a) Find Jack's expected net gain. Is this a fair game? b) What should Jack pay Bill as an entrance fee in order that the game to be fair? 28. Suppose two slot machines are available. Machine A costs $1 to play, with probability 1/3, returns $2 and with probability 2/3, returns nothing. Machine B also costs $1 to play but, with probability 1/6, return $4 and, with probability 5/6, return nothing. Consider the following two methods to play: 1). Play machine A first; if you win, play A again; otherwise switch to B. 2). Play machine B first; if you win, play B again; otherwise switch to A. a) Find the mean and variance of your net gain under the two possible methods of play. b) Which of the two methods would you prefer and why? Chapter 8 2. In three tosses of a balanced coin, let Z = number of tails W = number of heads a) Find E(Z), E(W), Var(Z), Var(W) b) Construct the joint p.f. of Z and W. Find Cov(Z, W). Are Z and W independent? c) Find the p.f. of Z+W. From this p.f., determine E(Z+W) and Var(Z+W) 3. A die is loaded so that the probability associated with a given face is proportional to the number on that face. Suppose such a die is rolled once. Let X = twice the number appearing on it Y = 1 if an odd number appears and 3 if an even number appears. a) Find the probability functions of X, Y and (X, Y) b) Determine E(XY), E(X), and E(Y) c) Are X and Y independent? 4. Let X and Y have the following joint probability function a) b) c) d) e) X/Y 1 2 3 1 0.1 0.1 0 2 0.1 0.2 0.3 3 0.1 0.1 0 Find the probability functions for X+Y, XY, X/Y Using the p.f.'s obtained in a), find E(X+Y), E(XY), E(X/Y), Var(X+Y) Find the p.f.'s of X and of Y. Determine E(X), E(Y), Var(X) and Var(Y) Using the results of b) and c), compare i) E(X+Y) and E(X) + E(Y) ii) E(XY) and E(X)E(Y) iii) E(X/Y) and E(X)/E(Y) Are X and Y independent random variables? Compare Var(X+Y) and Var(X) + Var(Y). Comment on this result. 6. In a freshman biology course students were given two term tests with the following results Text I, X1 60 10 Text II, X2 70 15 The instructor is considering two possible weighting schemes: Cov(X1,X2 ) 50 W1 = (1/2) X1 + (1/2) X2 W2 = (1/3) X1 + (2/3) X2 a) Find the mean and variance for each weighting scheme b) Which scheme would you prefer? 11. Consider two independent random variables X and Y with the following means and standard deviations: x = 50 ; x = 10 y = 60 ; y = 15 a) Find E(X+Y), Var(X+Y), E(X-Y), Var(X-Y) b) If X* and Y* are the standardized random variable's corresponding to the random variable's X and Y, respectively, determine E(X*+Y*), E(X*-Y*), Var(X*+Y*), Var(X*Y*). Chapter 9 1.Let X~b(x; n, p). a) For n = 6, p = 0.2, find (i) P(X>3), (ii) P(X3), (iii) P(X<2) b) For n = 15, p = 0.8, find (i) P(X2), (ii) P(X12), (iii) P(X=8) c) For n = 10, find p so that P(X8) = 0.6678 2. Let X be a binomial random variable with = 6 and 2 = 2.4. Find a) P(X>2) b) P(2