LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
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B.Sc. DEGREE EXAMINATION – STATISTICS
FIRST SEMESTER – November 2008
ST 1501 – PROBABILITY AND RANDOM VARIABLES
Date : 12-11-08 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
SECTION – A
Answer ALL the Questions (10×2=20 Marks)
- What do you mean by a random experiment?
- Using Axioms of Probability, show that P(Ac∩B) = P(B)-P(A∩B)
- A bag contains 3 white and 6 green balls. Another bag contains 6 white and 5 green balls. A ball is chosen from each bag. What is the probability that both will be green?
- If the letters of the word ‘REGULATION’ are arranged at random, what is the chance that there will be exactly 4 letters between R and E?
- A speaks truth in 60 percent cases and B in 70 percent cases. In what percentage of cases are they likely to contradict each other in stating the same fact?
- Define mutually independent events and pairwise independent events.
- The odds against Manager X settling the wage dispute with the workers are 8:6 and odds in favour of Manager Y settling the same dispute are 14:16. What is the chance that neither settles the dispute if they both try independently of each other?
- A problem in statistics is given to three students A, B and C, whose chances of solving it are 0.5, 0.75 and 0.25 respectively. What is the probability that the problem will be solved, if all of them try independently?
- Define Random Variable.
10.For the following probability distribution
- X : 1 2 3
- p(x): 1/2 1/3 1/6
Find variance of X.
SECTION – B
Answer any FIVE Questions (5×8 = 40 Marks)
- A committee of 4 people is to be appointed from 3 officers of production department, 4 officers of purchase department, 2 officers of sales department and 1 chartered accountant. Find the probability of forming the committee in the following manner
(a). There must be one from each department.
(b). It should have atleast one from the purchase department.
(c). A chartered accountant must be there in the committee. (2+4+2)
- (i). An MBA applies for a job in two firms X and Y. The probability of his being selected in the firm X is 0.7 and being rejected at Y is 0.5. The probability of atleast one of his applications being rejected is 0.6. What is the probability that he will be selected in one of the two firms?
(ii). If two dice are thrown, what is the probability that the sum is neither 7 nor 11? (4+4)
- For any three events A, B and C, Prove that
P(AUB/C) = PA/C)+P(B/C)-P(A∩B/C)
- Three groups of children contain respectively 3 girls and 1 boy, 2 girls and 2 boys and 1 girl and 3 boys. One child is selected at random from each group. What is the chance that three selected consists of one girl and two boys?
- A bag contains 17 tickets marked with the numbers 1 to 17. A ticket is drawn and replaced, a second drawing is made. What is the probability that (a) the first number drawn is even and the second is odd. (b) the first number is odd and the second is even? Find the corresponding probabilities under without replacement. (2+2+4)
- State and prove addition theorem on expectation when the random variable are continuous.
- State and prove Chebchev’s inequality.
- An urn contains four tickets marked with numbers 112, 121, 211, 222 and one ticket is drawn at random. Let Ai (i=1, 2, 3) be the event that ith digit of the number of the ticket drawn is 1. Discuss the independence of the events A1, A2 and A3.
SECTION – C
Answer any TWO Questions (2×20 = 40 Marks)
19.(i). State and prove Multiplication Theorem of Probability for n events.
(ii) The odds that a book on statistics will be favourably reviewed by three independent critics are 3 to 2, 4 to 3 and 2 to 3 respectively. What is the probability that of the three reviews: (a) all will be favourable (b) exactly two reviews will be favourable (c) atleast one of the reviews will be favourable. (10+10)
- (i). A and B play for a prize of Rs. 1000. A is to throw a die first and is to win if he throws 6. If he fails, B is to throw and is to win if he throws 6 or 5. If he fails, A is to throw again to win if he throws 6, 5 or 4 and so on. Find their respective expectations.
(ii). Two persons X and Y appear in an interview for two vacancies in the same post. The probability of X’s selection is 1/7 and that of Y’s selection is 1/5. What is the probability that (a) both of them will be selected (b) only one of them will be selected (c) none of them will be selected (d) atleast one of them will be selected? (10+10)
- (i). State and prove Baye’s Theorem.
(ii). A factory produces a certain type of output by three types of machines. The respective daily production values are
Machine I : 3000 units Machine II : 2500 units Machine III : 4500 units
Past experience shows that 1% of the output produced by machine I is defective, the corresponding fraction of defectives for other two machines are 1.2% and 2% respectively. An item is drawn at random from the day’s production run and is found to be defective. What is the probability that it comes from machine II. (10+10)
- An experiment consist of three independent tosses of a fair coin. Let X = The number of heads, Y = The number of head runs and Z = length of head runs. A head run being defined as consecutive occurrence of atleast two heads, its length being the number of heads occurring together in three tosses of the coin. Find the probability functions of (a) X (b) Y (c) Z. Also compute the mean and variance of the number of heads.
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