Name: Instructor: François Ledrappier

Math 323-01 Exam I, February 13, 2004

You may use your own calculator and your own textbook.

Part I consists of 6 multiple choice questions. Record your answers by placing an × through one letter for each problem on this answer sheet. Part II consists of 2 partial credit problem. Write your answer and show ALL your work on the page on which the question appears. Do not remove this answer sheet. You will return the whole exam. Each multiple choice question is worth 5 points. Each partial credit question is worth 15 points.

You are taking this exam under the honor code.

1. a  b  c  d  e

2. a  b  c  d  e

3. a  b  c  d  e

4. a  b  c  d  e

5. a  b  c  d  e

6. a  b  c  d  e

Multiple choice points:

Partial credit points:

Total points:

Part I: Multiple Choice Questions (5 points each)

1. Rank the following events from most likely to least likely: A. Two independent trials, each of which is a success with probability 0.7, all result in successes. B. Three independent trials, each of which is a success with probability 0.8, yield at least one failure. C. Seven independent trials, each of which is a success with probability 0.9, all result in successes.

(a) ABC  (b) ACB  (c) BAC  (d) CAB  (e) CBA

2. A cooling system can malfunction in two ways. It overheats with probability 0.15, and it leaks with probability 0.10. It automatically shuts down if both of these malfunctions happen at the same time. The probability that the system is running smoothly is 0.80. What is the probability that the system is running?

(a) 0.90  (b) 0.75  (c) 1.05  (d) 0.85  (e) 0.95

3. An urn contains 3 red and 2 black balls. A ball is taken randomly from the urn, its color is recorded, and the ball is put back into the urn. After five trials what is the probability that the number of red balls recorded is divisible by 2?

(a) \frac{1562}{3125}  (b) \frac{842}{3125}  (c) \frac{1472}{3125}  (d) \frac{3092}{3125}  (e) \frac{752}{3125}

4. Consider a grid in Figure 1. At any point you are only allowed to move one step to the right or one step up. Suppose that you start at the point A and go to the point B. How many different paths are there that do go through the point labelled X and do not go through the point labelled Y?

Figure 1

(a) 4,200  (b) 2,000  (c) 2,200  (d) 4,620  (e) 2,620

5. A pregnancy test is taken in the supposed second week of pregnancy. If the person taking the test is pregnant, then the test is positive with a .95 probability. If a woman taking the test is not pregnant, then the test is positive with a .05 probability. Assuming that at a given moment 0.5% of the women aged 20 to 35 are in the second week of a pregnancy, what is the probability for a woman in that age group of being pregnant if the test is positive?

(a)\frac{1}{2}  (b).488  (c).087  (d).219  (e).891

6. In Japan, you inadvertently knock over a display of six mineral gems. Unable to read Japanese, you hastily replace the gems and the labels at random. What is the probability that exactly two labels are placed near the corresponding gem?

(a) \frac{1}{3}  (b)\frac{3}{16}  (c)\frac{1}{15}  (d)\frac{121}{900}  (e) \frac{5}{24}

Part II: Partial Credit Problems (15 points each)

7.(15 points) An athlete is given three attempts to pass a qualifying test in order to advance to the next level in a competition. If he fails at i − 1 previous attempts, then the probability that he fails at his ith attempt is f_{i}. Given that f_{1} = 0.5, f_{2} = 0.4, and f_{3} = 0.7, find the probability that the athlete:

(a) passes at the second attempt;

(b) passes at the third attempt;

(c) passes given that he failed at the first attempt;

(d) passes at the second attempt given that he passes.

8.(15 points) You like playing tennis, and by experience, you know that among students your age and gender, there are 10% against whom you have a probability 0.2 of winning, 30% against whom your chances are even, and 60% against whom you have a probability 0.8 of winning. A new student arrives and you propose to test him or her by playing three independent games.

(a) The new student wins the first game. What is the probability that he or she is in the top category?

(b) The new student wins also the second game. What is the probability that he or she is in the top category?

(c) The new student then loses the third game. What is the probability that he or she is in the bottom category?