University of Connecticut Department of Mathematics Math 1070 Sample Exam 2 Fall 2014 Name: Instructor Name: Section: Exam 2 will cover Sections 4.6-4.7, 5.3-5.4, 6.1-6.4, and F.1-F.3. This sample exam is intended to be used as one of several resources to help you prepare. The coverage of topics is not exhaustive, and you should look through all examples from lectures, quizzes, and homework as these will all be relevant. The wealth of problems in our text is also a good resource for practice with this material. Read This First! Read the questions and any instructions carefully. The available points for each problem are given in brackets. You must show your work to obtain full credit (and to possibly receive partial credit). Calculators are allowed, but you must still show your work. Make sure your answers are clearly indicated, and cross out any work you do not want graded. If you finish early, check all your solutions before turning in your exam. Grading - For Administrative Use Only Page: 1 2 3 4 5 6 7 Total Points: 8 10 21 18 15 15 13 100 Score:
1. One of three bins is selected at random, each equally likely. From the chosen bin, a transistor is selected at random then tested. It is known that the first bin contains two defective and three non-defective transistors, the second bin contains four defective and three non-defective transistors, and the third bin contains only five defective transistors. (a) Draw a tree diagram for the given experiment, clearly indicating each probability. [4] (b) Given that a defective transistor was chosen, what is the probability that it came from [4] the second bin? Page 1 of 9
2. Three balls are randomly drawn (without replacement) from an urn that contains four white and six red balls. (a) Draw a tree diagram and indicate the correct probabilities. [3] (b) What is the probability of drawing a red ball on the third draw? [3] (c) What is the probability of drawing a red ball on the third draw given that at least one [4] red ball was drawn on the first two draws? Page 2 of 9
3. A basketball player makes on average 2 free throws out of every 3 attempted. If the player [6] attempts 6 free throws, find the probability that they make 4 or more of them. 4. Find the probability of a full house poker hand, that is, the number of poker hands with three [6] of a kind and two of another kind (eg. three Kings and two 8s, or three 5s and two Aces, etc.). 5. A baseball player has a batting average of 0.300 (this is the probability of getting a hit each time they bat). The player bats 3 times in a game. (a) What is the probability that the player gets exactly 2 hits? [4] (b) What is the player s expected number of hits? [5] Page 3 of 9
6. Three balls are selected at random from an urn that contains seven white balls and four red balls. Let the random variable X denote the number of white balls drawn. (a) Draw and complete a probability distribution table including all possible values of X. [8] (b) Draw a histogram for X. Make sure to label the axes and show all probabilities. [4] 7. A lottery has one $10,000 price, one $5,000 prize, three $1,000 prices, and ten $100 prices. [6] There are 10,000 lottery tickets sold at $3 each, and each is equally likely to win. Find the expected return on buying one lottery ticket. Page 4 of 9
8. The following is a list of the ages of the last eight presidents during their inaugurations: [6] 56, 61, 52, 69, 64, 46, 54, 47. Find the average, variance, and standard deviation. 9. The heights of players in a soccer league is normally distributed with mean µ = 71 inches and standard deviation σ = 5 inches. (a) Find the percentage of players that are less than 68 inches tall. [4] (b) Find the percentage of players that are more than 78 inches tall. [5] Page 5 of 9
10. A machine produces ball bearings with diameters normally distributed. The mean diameter is 3.50 cm, and the standard deviation is 0.02 cm. Quality requirements demand a ball bearing to be rejected if the diameter is more than 0.05 cm different from the mean. (a) Find P (X 3.48) where X is the diameter of a ball bearing. [4] (b) What is the probability of a ball bearing being rejected? [6] 11. Four cards are drawn from a standard 52-card deck. What is the probability that there are [5] two red cards and two spades? Page 6 of 9
12. Connecticut Direct Bank gives 4.25% interest compounded monthly. How much money did you [5] deposit five years ago if you have $10,000 in the account today? 13. Alex s parents want to send Alex to college. Alex s parents will make monthly deposits to their savings account, and aim to have $100,000 in 16 years time. (a) How much should Alex s parents save every month if the bank offers 5% interest compounded [5] monthly? (b) How much total interest does this annuity earn in 16 years? [3] Page 7 of 9
Simple Interest F = P (1 + rt) r eff = r 1 rt Compound Interest ( F = P 1 + m) r mt = P (1 + i) n ( r eff = 1 + m) r m 1 = (1 + i) m 1 Future Value of Annuities FV = PMT (1 + i)n 1 i Present Value of Annuities PV = PMT 1 (1 + i) n i Formulas From Chapter F Page 8 of 9
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