MATH171501
This MATH171501 question paper consists of 5 printed pages, each of which is identied by the reference MATH171501. Statistical tables are attached at the end of the question paper. Only approved basic scientic calculators may be used.
c UNIVERSITY OF LEEDS Examination for the Module MATH1715 (January 2008) INTRODUCTION TO PROBABILITY Time allowed: 2 hours Attempt ALL questions in Section A and THREE questions from Section B. For Section A only write down a single letter answer for each question. Section A is worth 40% of the examination marks. All questions within each section carry equal marks.
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MATH171501
Section A
Attempt ALL questions in Section A. For each question, write down a single letter answer. A1. An integer is selected at random from the set {1, 2, . . . , 1 000 000}. What is the probability that it contains the digit 5? A: 0.5 B: 0.531 C: 0.6 D: 0.04 E: 0.469
A2. If events A and B are mutually exclusive then it is always true that A: P(A B) = P(A) P(B) B: P(A) + P(B) = 1 C: P(A B) = P(A) + P(B)
D: P(A \ B) = P(A) P(B)
E: P(A B c ) = P(Ac B)
A3. A contractor has 8 suppliers from which to purchase electrical supplies. He will select 3 of these at random and ask each supplier to submit a project bid. If your rm is one of the 8 suppliers, what is the probability that you will get the opportunity to bid on the project? A: 0.125 B: 0.375 C: 0.250 D: 0.5 E: 0.568
A4. Two university students were partying before a test, and the next morning they didnt get to the university until the test was over. Their excuse to the lecturer was that they were on a car and had a at tyre, so they asked if they could take a make-up test. The lecturer agreed, wrote out a test and sent the two to separate rooms to take it. On one side of the paper, there was one question worth 20 points, and they answered it easily. Then they turned the paper over and found the second question, worth 80 points: What tyre was it? What is the probability that both students would say the same thing? 1 1 1 2 1 A: B: C: D: E: 16 8 4 5 2 A5. Suppose that the waiting time in a queue has an exponential distribution with mean 20 minutes. What is the probability that standing in the queue will take no more that 20 minutes? A: 0.736 B: 0.264 C: 0.5 D: 0.632 E: 0.368
A6. We toss a coin repeatedly until we see each side of the coin at least once. What is the mean number of tosses required? A: 1.5 B: 2 C: 2.5 D: 3 E: 4
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MATH171501 A7. An insurance company writes a policy to the effect that an amount of 1000 must be paid if a certain event A occurs within a year. The company estimates that A will occur within a year with probability 0.02. What should be the premium c (i.e., the amount in pounds the customer is charged when buying the policy) in order that the companys expected prot per policy be 50% of c? A: 50 B: 40 C: 70 D: 60 E: 30
A8. Suppose there are n ceramic money-boxes, each having a lock with a different key. The n keys are dropped at random into the boxes (one key per box). You are allowed to smash one box at your choice. Using the key from that box, you may be able to open another box and get a new key, which may open the next box and so on. You continue this way to open as many boxes as possible. What is the probability that you will manage to get all n keys? A: 1 n B: 2 n+1 C: 1 n! D: 1 nn E: n! nn
A9. If events A and B have probabilities 0.7 and 0.3, respectively, and P(A B) = 0.8, what is the probability that both events A and B occur? A: 0.4 B: 0.2 C: 0.21 D: 0.8 E: 1
A10. A football team loses each game with probability 0.4, independently of other games. What is the probability the team will lose two games out of six? A: 0.544 B: 0.160 C: 0.138 D: 0.311 E: 0.261
Section B
Attempt THREE questions from Section B.
B1.
(a) Let A, B , and C be arbitrary events. Using Venn diagrams, determine if the following equality is correct: (A B) \ C = (A \ C) (B \ C). (b) Let X1 , X2 and X3 be independent random variables with zero mean and variance 1. Find the correlation coefcient between Y1 = 2X1 + X2 and Y2 = X2 2X3 . (c) A balanced die is rolled 180 times. Let X be the number of rolls where the die shows a 6. Use the normal approximation with continuity correction to evaluate the probability that X = 30.
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MATH171501 (a) B2. For a Democratic candidate to win an election, she must win the districts I, II, and III. Polls have shown that the probability of winning I and III is 0.55, losing II but winning I is 0.34, and losing II and III but not I is 0.15. Find the probability that the candidate will win all three districts. (b) Two hunters shoot at a duck, which is hit by exactly one bullet. If the rst hunter hits his target with probability 0.3 and the second with probability 0.6, what is the probability that the second hunter killed the duck? (c) For a car travelling at 30 miles per hour (mph), the distance required to brake to a stop is normally distributed with mean 50 feet and a standard deviation 8 feet. Suppose you are driving at 30 mph in a residential area and a car in front of you stops abruptly at a distance of 60 feet. If you brake immediately to a stop, what is the probability that you will avoid the collision? B3. (a) The probability that a certain type of inoculation takes effect is 0.995. Using the Poisson approximation or otherwise, compute the probability (to 3 decimal places) that at most 2 out of 400 people given the inoculation will nd that it has not taken effect. (b) In the EuroMillions lottery, a player must choose ve different numbers from 1 to 50 on the main board and two lucky stars from nine digits 1, . . . , 9. The lottery draw yields a 5 + 2 winning combination, that is, ve numbers on the main board and two lucky stars (the order in which they appear in the draw is not important). What is the probability that the player will guess correctly two numbers (out of ve) on the main board and one lucky star (out of two)? Give the answer to 3 signicant gures. (c) Three factories manufacture 20%, 30% and 50% of computer chips a company sells. If the fractions of defective chips are 0.4%, 0.3% and 0.2%, respectively, what fraction of the defective chips come from the third factory? Give the answer to 3 signicant gures. B4. (a) A box contains three blue balls and ve red balls. Ann and Bob take turns (with Ann going rst) to draw a ball from the box, without replacement, until a blue ball is drawn. What is the probability that Ann will draw the blue ball? (b) Suppose that random variables X and Y are nonnegative and their joint cumulative xy distribution function is given by FXY (x, y) = for x, y 0. (x + 1)(y + 1) (i) Compute the probability P{1 X 3, 1 Y 4}. (ii) Obtain the marginal probability density function of the random variable X . (c) Suppose that a random variable X has exponential distribution with pa...