LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AB 02

FIRST SEMESTER – NOV 2006

ST 1501 – PROBABILITY AND RANDOM VARIABLES

 

 

Date & Time : 03-11-2006/1.00-4.00     Dept. No.                                                       Max. : 100 Marks

 

 

SECTION – A

Answer ALL the Questions                                                                    (10 x 2 = 20 marks)

 

1. Define Mutually Exclusive Events and give an example.
2. Using the Axioms of Probability, prove that if AB, P(A) ≤ P(B).
3. Draw Venn Diagram to represent the occurrence of at least two of three events A, B, C.
4. State the number of ways in which a set of 10 objects can be partitioned into three subsets of sizes 5, 3 and 2 respectively.
5. Explain the ‘Matching Problem’ and find the probability that a specified match occurs.
6. In an experiment of forming three-letter English words with three distinct letters, what is the probability that the first and third letters are consonants while the second one is a vowel?
7. Four statisticians arrange to meet at Hotel Five Star in a city. But there are four hotels in the city with the same name. If each one randomly chooses a hotel, what is the probability that they will all choose different hotels?
8. In tossing a coin thrice, what is the conditional probability that the first toss results in Head given that there are two Heads?
9. If A and B are independent events, show that A and B^c are independent.
10. Define a Continuous random variable.

SECTION – B

 

 

Answer any FIVE Questions                                                                    (5 x 8 = 40 marks)

 

1. State the Binomial Theorem. Using Pascal’s Triangle write down the expansion of (a + b)⁶.

 

1. A study of 1000 people revealed that 500 were successful in their careers, 300 had studied Statistics and 200 had both studied Statistics and were successful in their careers. Find the probability that a randomly chosen person from the lot

• neither studied Statistics nor is successful in his career.
• is successful in his career but had not studied Statistics.
• had studied Statistics but is not successful in his career.

 

1. Two fair dice are thrown. Find the probability that (a) at least one of the dice shows up an odd number; (b) the sum of the two numbers exceeds 7.           (4 + 4)

 

1. In a random rearrangement of the letters of the word COMMERCE, find the probability that

• All the vowels come together
• All the vowels occupy odd number positions. (4 + 4 )

 

1. Show that P(A | B ) + P (A^c | B) = 1.  Give an example to show that in general  P(A | B) + P(A | B^c) ≠ 1.

 

1. An urn contains 12 balls out of which 8 are white. A sample of size 4 is drawn one by one. Find the conditional probability that the first ball drawn is white given that the sample contained two white balls. Solve this under with replacement and without replacement selections.

 

 

 

 

 

1. A number is drawn at random from 1 to 4. Discuss the independence of the following three events:

A: Number is < 3,  B: Number is odd,  C: Number is 1 or 4

 

1. State and prove the ‘Multiplication Theorem of Probability’ for many events.

 

SECTION – C

 

 

Answer any TWO Questions                                                                 (2 x 20 = 40 marks)

 

1. (a) State the Addition Theorem of Probability for two events. Hence state and prove the same for three events.

(b) Let A and B be events. For k = 0, 1, 2, express the following probabilities in

terms of P(A), P(B) and P(AB):

• P( exactly ‘k’ of the events A and B occur)
• P(at least ‘k’ of the events A and B occur) (8 +12)

 

1. An urn contains 5 white and 4 black balls from which 3 balls are drawn. Find the probability that (i) all three are black, (ii) at least two are white. Solve this under with replacement, without replacement and subset (unordered) selection schemes.

 

1. (a) State and prove the ‘Law of Total Probability’. Hence establish Baye’s Theorem.

(b) Three persons A₁, A₂, A₃ compete for the position of finance Managers of a company. The probability for A₁ to get appointed is 3/8, for A₂ it is 1/2 and for A₃ it is 1/8. If A₁ becomes the manager, the probability that he will introduce a Bonus scheme is 3/10. The corresponding probabilities in the case of A₂ and A₃ are 1/3 and 3/5. Given that the Bonus scheme has been introduced, what is the probability that A₁ was appointed?                                                                                 (10 +10)

 

1. (a) A random variable X has the following p.m.f.:

x	-2	-1	0	1	2
P(X =x)	1/10	1/5	2/5	1/5	1/10

 

 

 

Find the c.d.f. of X. Also. Find the mean and variance of X.

(b) If f(x) = cx², – 1< x < 2 , is the p.d.f. of a continuous random variable X, find

the value of ‘c’ and evaluate P( 0 < X < 1).                                                (13 +7)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

AC 02

DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2007

ST 1501 – PROBABILITY AND RANDOM VARIABLES

 

 

Date & Time: 26/04/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

 

 

 

 

SECTION – A

Answer ALL the Questions                                                                    (10 x 2 = 20 marks)

1. Define Random Experiment and Sample Space.
2. Using the Axioms of Probability, show that P(A^c) = 1 – P(A).
3. Draw a Venn diagram to represent the occurrence of exactly two of three events A, B, C.
4. State the exhaustive number of ways of choosing 2 balls one-by-one from a collection of 5 balls (i) with replacement, (ii) without replacement.
5. A restaurant menu lists 3 soups, 10 rice varieties, 5 desserts and 3 beverages. In how many ways can a meal consisting of all the four be ordered?
6. How many ‘distinct words’ can be formed from the letters of the word MISSISSIPPI.
7. Find the probability that number 2 shows up in one of the two dice thrown given that the sum of the two numbers got is 7.
8. Two persons A and B can independently solve a problem in mathematics with probabilities 1/5 and 1/3 respectively. If a problem is posed to them, what is the probability that the problem will be solved?
9. If A and B are independent events show that A^c and B are independent.
10. Define a Random Variable.

 

SECTION – B

 

 

Answer any FIVE Questions                                                                    (5 x 8 = 40 marks)

1. Using the Axioms of Probability, prove the following:

(a) If AB, P(A) ≤ P(B)

(b) P(A – B) = P( A ) –  P(AB)                 (4 + 4)

1. State the Binomial Theorem. Using Pascal’s Triangle, write down the expansion of  (a + b )⁵.

(4 + 4)

 

1. An urn contains 3 red balls, 4 white balls and 5 blue balls. Another urn contains 5 red, 6 white and 7 blue balls. One ball is selected from each urn. Find the probability that (i) blue and red balls are selected; (ii) the two are of different colours.                                       (3 + 5)

 

1. A box contains 10 tickets numbered 0 to 9 and from it three are chosen one by one. By placing the numbers in a row, an integer between 0 and 999 is formed. What is the probability that the integer so formed is divisible by 39 (regarding 0 as divisible by 39). Solve this under ( i) with replacement, (ii) without replacement, sampling schemes.

 

1. A man tosses two fair dice. What are the conditional probabilities that the sum of the two dice is 7 given that (i) the sum is odd? (ii) the sum is greater than 5?

 

1. Two fair dice are thrown. Discuss the independence of the following three events:

A: First die shows odd number

B: Second die shows odd number

C: Sum of the two numbers is odd

 

 

 

1. Consider two events A and B with P(A) = ¼. P(B| A) = ½, P( A| B) = ¼. Verify whether the following statements are true:

(i) A is a sub-event of B; (ii) P(A | B) + P( A| B^c) = 1

 

1. State and prove the ‘Multiplication Theorem of Probability’ for many events.

 

SECTION – C

 

 

Answer any TWO Questions                                                                 (2 x 20 = 40 marks)

 

19.(a) State and prove the Addition Law of Probability for two events. State the

extension for three events.

(b) Consider the following statements about the subscribers of a magazine with

respect to their gender, marital status and education:

P(Male) =0.312, P(married) = 0.470, P(Graduate) = 0.525, P(Male Graduate) = 0.042,                                                                   P(Married Graduate) = 0.147, P(Married Male) = 0.086, P(Married male graduate)= 0.025.

Show that the information is wrong.

 

• There are three urns with the following contents.

Urn I:    3 white, 2 red, 5 black balls

Urn II:   4 white, 1 red, 5 black balls

Urn III:  4 white, 4 red, 2 black balls.

One ball is chosen from each urn. Find the probability that in the sample drawn  (a) there are exactly 2 black balls, (b) Balls of any two colours are found.   (8 +12)

 

1. (a) State and prove the “Law of Total Probability’. Hence establish Baye’s Theorem.

(b) Three Companies X, Y, Z manufacture tube lights. The market shares of the companies are 50% for X, 30% for Y and 20% for Z. 5% of the tubes manufactured by X are defective, 1% from Y and 2% from Z are defective. A bulb is chosen at random and is found to be defective. What is the probability that it was manufactured by Z?                                                                           (10 +10)

 

1. A loaded coin with Heads twice as likely as Tails in any toss is tossed thrice. Write down the sample space of the experiment. Obtain the p.m.f. and hence the c.d.f. of the number of Heads. Also, compute the mean and variance of the number of Heads.

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 2

           B.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2008

ST 1501 (PROBABILITY AND RANDOM VARIABLES)

 

 

 

Date : 06-05-08                  Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION –A

 

Answer ALL the questions.                                                                         (10×2=20 marks)

 

1. State the mathematical definition of probability.
2. A, B and C are three mutually exclusive and exhaustive events associated with a random experiment. Find given .
3. If A, B and C are three arbitrary events, find expressions for the events given below:

1. Both A and B but not C occur
2. At least one occur

1.  Two dice are tossed. Find the probability of getting an even number on the first die.
2. For any event , show that A and null even are independent.
3. The odds in favour of manager X settling the wage dispute with the workers are 6:8 while the odds in favour of manager Y settling the same dispute are 14:16. What is the probability that the dispute is settled?
4. Define ‘aprior’ and ‘posterior’ probabilities.
5. A university has to select an examiner from a list of 50 persons – 20 of them being women and 30 men, 10 of them know Tamil and 40 not, 15 of them are teachers while the remaining 35 are not. What is the probability that the University selects a Tamil knowing woman teacher?
6. Define a random variable.
7. If X is a random variable and ‘a’, ‘b’ are constants, then show that .

 

SECTION –B

Answer any FIVE questions.                                                           (5×8=40 marks)

 

1.  If , then show that

 

1. Twenty five books are placed at random in a shelf. Find the probability that a particular pair of books shall be (i) always together and (ii) never together.
2. If A₁, A₂, …, A_n are ‘n’ events, then show that
3. Three newspapers A, B and C are published in a certain city. It is estimated from a survey that, out of the adult population 20% read A, 16% read B, 14% read C, 8% read both A and B, 5% read both A and C, 4% read both B and C, 2% read all the three. Find what percentage read atleast one of the papers.
4. An urn contains four tickets marked with numbers 112, 121, 211, 222 and one ticket is drawn at random. Let A_i , i=1,2,3 be the event that the i^th digit of the number of the ticket drawn is 1. Discuss the independence of the events A₁, A₂ and A₃.

 

 

1. A random variable x has the following probability mass function:

x:	0	1	2	3	4	5	6	7
p(x):	0	k	2k	2k	3k	k2	2k2	7k2+k

1. i) Find k ii) Evaluate

1. Let X be a random variable with the following probability distribution:

x:	-3	6	9
p(x)	1/6	1/2	1/3

Find and hence .

1. State and prove multiplication theorem of probability for ‘n’ events.

SECTION –C

 

Answer any TWO questions.                                                           (2×20=40 marks)

 

1. a) State and prove addition theorem of probability for 3 events.       (10)

1. b) Three groups of children contain 3 girls and 1 boy; 2 girls and 2 boys; 1 girl and 3 boys respectively. One child is selected at random from each group. Find the chance that the three selected consist of 1 girl and 2 boys. (10)

1. a) State and prove Baye’s theorem for future events.

1. b) The probability of x, y and z becoming managers are The probability that the Bonus scheme is introduced if x, y and z become mangers are , and respectively. i) What is the probability that the Bonus scheme will be introduced? ii) If the Bonus scheme has been introduced, what is the probability that the manager appointed was x?

1. Two dice, one green and the other red are thrown. Let A be the event that the sum of the points on the faces is odd and B be the event that atleast one number is 1.

1. Define the complete sample space and the events A, B, and
2. Find the probability of the events:

, ,,

1. a) State and prove chebychev’s inequality

1. b) Let the random variable x have the distribution:

.

For what value of p is the var (x) a maximum?

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

BA 02

   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)

 

1. What do you mean by a random experiment?
2. Using Axioms of Probability, show that P(A^c∩B) = P(B)-P(A∩B)
3. 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?
4. 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?
5. 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?
6. Define mutually independent events and pairwise independent events.
7. 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?
8. 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?
9. Define Random Variable.

10.For the following probability distribution

1. X : 1          2          3
2. p(x): 1/2       1/3       1/6

Find variance of X.

 

SECTION – B

Answer any FIVE Questions                                                             (5×8 = 40 Marks)

 

11. 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)

12. (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)

13. For any three events A, B and C, Prove that

P(AUB/C) = PA/C)+P(B/C)-P(A∩B/C)

14. 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?
15. 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)
16. State and prove addition theorem on expectation when the random variable are continuous.
17. State and prove Chebchev’s inequality.
18. An urn contains four tickets marked with numbers 112, 121, 211, 222 and one ticket is drawn at random. Let A_i (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)

20. (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)

21. (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)

22. 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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

YB 02

FIRST SEMESTER – April 2009

ST 1503/ ST 1501 – PROBABILITY AND RANDOM VARIABLES

 

 

 

Date & Time: 22/04/2009 / 1:00 – 4:00         Dept. No.                                                       Max. : 100 Marks

 

 

PART – A

Answer ALL the questions:                                                             (10 x 2 = 20)

1. Define sample space and elementary event.
2. Four cards are drawn at random from a pack of 5 cards. Find the probability that two are kings and two are queens.
3. State the axioms of probability function.
4. For any two events A and B, show that .
5. Define pairwise and mutual independence of events.
6. The odds that a book in mathematics will be favourably reviewed by 3 independent critics are 3 to 2, 4 to 3 and 2 to 3 respectively. What is the probability that all the three reviews will be favourable?
7. If A and B are two mutually exclusive events show that .
8. A consignment of 15 record players contains 4 defectives. The record players are selected at random, one by one and examined. Those examined are not replaced. What is the probability that the 9^th one examined is the last defective?
9. Define a random variable.
10. Define probability generating function.

PART – B

Answer any FIVE questions:                                                                       (5 x 8 = 40)

11. Five salesmen A, B, C, D and E of a company are considered for a three member trade delegation to represent the company in an international trade conference. Construct the sample space and find the probability that i) A is selected ii) either A or B is selected.
12. For any three non-mutually exclusive events A, B and C, evaluate .
13. Compare the chances of throwing 4 with one die, 8 with two dice and 12 with three dice.
14. State and prove Boole’s inequality.
15. A and B are two students whose chances of solving a problem in statistics correctly are and If the probability of them making a common error is and they obtain the same answer, find the probability that their answer is correct.
16. Let the random variable X have the distribution þþ where þ. For what value of þ is the variance of X, a maximum.
17. The life (in 000’s of kms) of a car tyre is a random variable having the p.d.f.

Find the probability that one of these tyres will last i) at most 10,000 kms ii) at least 30,000 kms.

18. If X and Y are two random variables such that , then show that .

 

 

 

 

PART – C

Answer any TWO questions:                                                                       (2 x 20 = 40)

19. (a) State and prove the multiplication theorem of probability for any ‘n’ events.

(b) In a company, 60% of the employees are graduates. Of these 10% are in sales. Out of the employees who are not graduates, 80% are in sales. What is the probability that

1. i) an employee selected at random is in sales
2. ii) an employee selected at random is neither in sales nor a graduate?
3. a) State and prove Baye’s Theorem.
4. b) Two computers A and B are to be marketed. A salesman who is assigned the job of finding customers for them has 60% and 40% chances respectively of succeeding in case of computers A and B. The two computers can be sold independently. What is the probability that computer A is sold given that he was able to sell atleast one computer?
5. a) State and prove chebychev’s inequality
6. b) A symmetric die is thrown 600 times. Find the lower bound for the probability of getting 80 to 120 sixes.
7. The time a person has to wait for a bus is a random variable with the following p.d.f.

Let the event A be defined as a person waiting between 0 and 2 minutes (inclusive) and B be the event of waiting 0 and 3 minutes (inclusive)

1. Draw the graph of p.d.f
2. Show that and .

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – NOVEMBER 2010

ST 1503/ST 1501 – PROBABILITY AND RANDOM VARIABLES

 

 

 

Date : 12-11-10                     Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

 

Answer ALL the questions:                                                                                          (10 x 2 =20 marks)

 

1. What do you mean by “Random Experiments”?
2. A letter of the English alphabet is chosen at random. Calculate the probability the letter

so chosen (i) is a vowel ;  (ii) precedes J and is a vowel.

3. State Classical definition of probability.
4. All cards of ace, jack and queen are removed from a deck of playing cards.  One card is

drawn at random from the remaining cards.  Find the probability that the card drawn is

• a face card and    (ii) not a face card.

5. If P(A) = 0.3, P(B) = 0.2, P(C) = 0.1 and A,B,C are independent events, find the

probability  of occurrence of at least one of the three events A,B, and C.

6. Two coins are tossed. Show that the event “ head on first coin “ and event “ Coins fall alike”

are independent.

7. A person is known to hit a target in 5 out of 8 shots, whereas another person is known to

hit in 3 out of 5 shots. Find the probability that the target is hit at all when they both try.

8. If for three mutually exclusive and exhaustive events A,B and C;  and

P(B) = 2/3 P(C) then find P(A).

9.  Find the standard deviation of the probability distribution

x = x	0	1	2
P(x)

 

10. Find the mathematical expectation of the number of points if a balanced die is thrown.

 

PART – B

Answer any FIVE questions                                                                                 (5 x 8 = 40 Marks)

 

11. For three non-mutually exclusive events A,B, and C, prove that

P( A B C ) = P(A) +P(B)+P(C) – P(A B) – P( AC) – P(BC ) + P(A B C).

 

 

12. Prove that

• P( B ) = P(B) – P(A B)
• P(A ) =  P(A) – (A B).

13. In a random arrangement of the letters of the word “MATHEMATICS”, find the

probability that all the vowels come together.

14. If events A and B are independent then prove that the complementary events and are

also independent.  Also show that A and   are independent.

15. An urn contains four tickets marked with numbers 112,121, 211, 222 and one ticket is

drawn at random.  Let A_i ( 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.

16. State and prove multiplication law of probability when the events are (i ) not independent

(ii) independent.

 

17. Let X be a continuous random variable with p.d.f given by

f(x)   =  kx   ,                    0 £ x £ 1

=  k   ,                     1 £ x £ 2

=   kx + 3k   ,        2 £ x £ 3

=   0   , otherwise

1. Determine the constant k
2. Determine F(x).
3. A continuous random variable X has the following p.d.f

f(x) = 3 x²  ,      0 < x < 1

=  0  ,         otherwise

Verify that it is a p.d.f and evaluate the following probabilities

1. P( X 1/3 )    (ii)  P ( 1/3  X  ½ )  and  (iii)  P (X  1/2)  1/3  x  2/3 ).

 

PART – C

 

Answer any TWO questions                                                                                     (2 x 20 = 40 Marks)

19. a) Let A and B be two possible outcomes of an experiment and suppose P(A) =0.4,

P( A B) = 0.7 and P(B) = p

• For what choice of p, are A and B mutually exclusive?
• For what choice of p, are A and B independent?

 

1. b) There are two bags. The first contains 2 red and 1 white ball, whereas the second bag

has only 1 red and 2 white balls. One ball is taken out at random from the first bag and

put in the second.  Then a ball is chosen at random from the second bag. What is the

probability that this last ball is red ?

20. a) A and B play 12 games of chess of which 6 are won by A, 4 by B and 2 end in a tie.

They agree to play 3 more games.  Find the probability that (i) A wins all the three

games (ii) two games end in a tie (iii) A and B win alternatively and (iv) B wins at least

one game.

 

1. b) For n events A₁, A₂ … A_n , Prove that

(i)  P (A_i) – (n-1)

(ii) P (A_i).

21. a) State and prove Baye’s Theorem.

 

1.     b) There are ‘n’ boxes each containing 4 white and 7 black balls.  Another one box has got 7

white and 4 black balls.  A box is selected at random from the (n + 1 ) boxes and 2 balls

are drawn out of it and both are found to be black.  If it is now calculated that probability

that there are 7 white and 2 black balls remaining in  the chosen box is 1/ 15.  What is the

value of ‘n’?

 

22. a) The length of time ( in minutes) that a certain lady speaks on the telephone is found to be

random phenomenon with a probability function specified by the probability density

function f(x) as

f(x) = A e^{– x/5}     for x  0

=  0,      otherwise

• Find the value of A that make a p.d.f
• What is the probability that the number of minutes that she will talk over the phone is
• more than 10 minutes
• less than 5 minutes
• between 5 and 10 minutes?

 

1. b) State and prove Chebyshev’s inequality?

 

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2011

ST 1503/ST 1501 – PROBABILITY AND RANDOM VARIABLES

 

 

 

Date : 06-04-2011             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

Section A                                Answer ALL the questions                                      10×2=20 

 

1. Define mutually Exclusive events with an example.
2. Mention any two limitations of classical definition of probability.
3. What is the probability that a leap year selected at random will have 53 Sundays?
4. When do we say that an event A is statistically independent with respect to an event B?

Does this mean that event B is statistically independent of the event A?

5. If A and B are independent events, Show that A^c and B are independent events.
6. What is the importance of the Baye’s Theorem?
7. Explain the term Bernoulli trials.
8. If X is a continuous random variable with probability density function f(x)      =  k x (1-x),

if  0 £          x £ 1    and 0, otherwise, find the value of k. Hence find E (x).

9. If X and Y are two random variables such that X £ Y, show that

E(X) £ E(Y) provided they exist.

10. State any two properties of variance of a random variable.

Section B                                Answer any FIVE questions                                                5×8=40 

 

11. Define Probability Generating Function. (PGF). Obtain the PGF of a random variable X that

follows a Poisson distribution with parameter, l. Hence find the mean and variance of X.

12. A box contains 6 red, 4 white and 5 black balls. A person draws 4 balls from the box at

random. Find the probability that among the balls drawn, there is at least one ball of each colour.

13. Using axioms of probability show that, for any two events A and

B,            P (AÈB) + P (AÇB) = P (A)+P (B)

14. From a vessel containing 3 white and 5 black balls, 4 balls are transferred into an empty vessel.

From this vessel a ball is drawn and found to be white. What is the probability that, out of four

balls transferred, 3 were white and 1 was black?

15. A bag contains 10 gold and 8 silver coins. Two successive drawings of 4 coins are made such

that (i) coins are replaced before the second trial, (ii)  the coins are not replaced before the

second trial. Find the probability that the first drawing will give 4 gold and the second 4 silver

coins.

 

 

 

 

16. Sixty per cent of the employees of the XYZ Corporation are college graduates. Of these, ten per

cent are in sales. Of the employees who did not graduate from the college, eighty per cent are

in sales.

• What is the probability that an employee selected at random is in sales? (b) What is the

probability that an employee selected at random is neither in sales nor a college graduate?

17. The odds against Manager X settling a wage dispute with the workers are 8: 6. The odds in

favour  of Manager Y settling the same dispute are 14:16. (a) What is the probability that

neither settles the dispute, if they both try independently of each other? (b) What is the

probability that the dispute will be solved?

18. If X is a continuous random variable with probability density function, f(x) = 1 if 0 < x < 1 and

0, otherwise, Use Chebyshev’s Inequality   to obtain an upper bound for P [ |X –E(X) | > 2s ].

Compare it with the exact probability.

 

Section C                                Answer any TWO questions                                                            2×20=40 

 

19. (a) If A, B and C are mutually independent show that AÈB and C are independent

(b) An Urn contains four tickets marked with numbers, 112, 121, 211, 222, and one ticket is

drawn at random. Let A_i (i=1, 2, 3, ) be the event that ith digit of the number of the ticket

drawn is 1. Discuss the mutual independence and pair-wise independence of the events.

 

20. (a) State and prove Bayes theorem for future events.

(b)   Suppose that Urn I contains 1 white, 2 black and 3 red balls. Urn II contains 2 white, I black

and 1 red balls. Urn III contains, 4 white, 5 black and 3 red balls. One Urn is chosen at

random and two balls are drawn from it. They happen to be white and red. What is the

probability that they have come from I, II or III?

21. (a) State and prove Chebyshev’s inequality. Bring out its importance.

(b) A petrol pump is supplied with petrol once in a day. If its daily volume of sales (X) (in

thousands of litres) is distributed by

f(x)     =  5 (1-x) ⁴,                  if  0 £  x £       1

What must the capacity be of is tank in order that the probability that its supply will get

exhausted in a given day shall be 0.01?

22. (a) For any two events A. B and C, show that

P (AÈB÷ C )  = P (A÷ C) + P (B÷ C) – P (AÇB÷ C)

and     P (AÇB^C÷ C )  + P (AÇB÷ C) = P (A÷ C).

• Two computers A and B are to be marketed. A salesman who was assigned the job of finding customers for them has 60% chance of succeeding in the case of computer A and 40% chance of succeeding in the case of Computer B. The two computers can be sold independently. Given that he was able to sell at least one computer, what is the probability that computer A has been sold.

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2012

ST 1503/ST 1501 – PROBABILITY AND RANDOM VARIABLES

 

 

 

Date : 02-05-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART –A

Answer ALL the questions:                                                                                     ( 10 x 2 = 20 Marks )

 

1. Distinguish between Mutually Exclusive event and Independent event.
2. Prove that for any event A in S, P( A ∩ A^C ) = 0.
3. Suppose from a pack of 52 cards one card is drawn at random, what is the probability that it is either a king or a queen.
4. A fair coin is tossed 5 times. What is the probability of having at least one head?
5. Prove that if P(A) > P(B), then P(A│B) > P(B│A)
6. If two events A and B are independent, show that (i) A^C and B^C are independent.
7. Out of 800 families with 4 children each, how many families would be expected to have at least one boy?
8. A bag contains 5 white and 3 black balls.  Two balls are drawn at random one after the other without replacement.  Find the probability that both balls drawn are black.
9. A fair coin is tossed three times.  Let X be the number of tails appearing.  Find the probability distribution of X .  Calculate E(X).
10. State addition theorem of probability for two events A and B.

PART –B

Answer any FIVE questions:                                                                                    (5 x 8 = 40 Marks)

 

11. Given P(A) = 1/ 3, P(B) = 1/ 4 and P (A ∩ B ) = 1 / 6

Find the following probabilities  (i) P ( A^C ) , (ii) P ( A^C  ÈB ) and (iii) P ( A^C ∩ B^C ).

12 . (a) An MBA candidate applies for a job in two firms X and Y.  The probability of his being selected

in firm X is 0.7 and being rejected at Y is 0.5.  The probability of at least one of his

applications being rejected is 0.6.  What is the probability that he will be selected in one

of the firms?

(b) What is the chance that a leap year selected at random will contain 53 Sundays?

13 (a) A pair of dice is rolled.  If the sum of  9 has appeared, find the probability that one of the

dice shows 3.

(b) Two a’s and b’s are arranged in order.  All arrangements are equally likely. Given that the

last letter, in order is b’ find the probability that the two a’s are together.

1. Two urns contain 4 white and 6 black balls and 4 white and 8 black balls.  One urn is selected at random and a ball is taken out.  It turns out to be white.  Find the probability that it is from the first urn.
2. It is given that P( AÈB ) = 5 /6 , P ( A ∩B ) = 1 /3 and P ( B^C ) = 1/ 2.  Show that the

events A and B are independent.

16. Let X be a continuous random variable with p.d.f given by

f(x) = K x         ,       0 ≤ x < 1

= K            ,       1 ≤ x < 2

= ─ Kx + 3 K  ,  2 ≤ x ≤ 3

=  Otherwise

• Determine the constant K (ii)  Determine F(x)
• State and prove Multiplication law of probability.
• In four tosses of a coin, let X be the number of heads. Tabulate the 16 possible outcomes with the corresponding values of X. By simple counting, derive the probability distribution of X and hence calculate the expected value of X.

SECTION – C

 Answer any TWO questions:                                                                               (2 x 20 = 40 Marks)

• a) Three groups of children contain respectively 3 girls and 1 boy and 2 girls and 2 boys and 1 girl

and 3 boys.  One child is selected at random from each group.  Find the chance that the 3

selected comprise 1 girl and 2 boys.

1. b) A, B and C go for bird hunting . A has record of 1 bird out of 2, B gets 2 out of 3 and C gets 3

out of 4.  What is the probability that they will kill a bird at which all shoot simultaneously?

• a) An unbiased coin is tossed three times. Let A be the event “ not more than one head” ,

and let B be the event “ at least one of each face “.  Are A and B independent?

1. b) Two persons A and B attempt independently to solve a puzzle. The probability that A will

solve it is 3/5 and the probability that B will solve it is 1/3.  Find the probability that the

puzzle will be solved by (i) at least one of them and (ii) both of them

21.   a) State and prove Baye’s Theorem.
22. b) If A, B, C are mutually independent events then prove that and C are also independent.
23. The length of time ( in mintues) that a certain lady speaks on the telephone is found to be

random phenomenon, with a probability function specified by the probability density

function f(x) as

f(x) = A e  ^{– X / 5}   , for x ≥ 0

=  0 otherwise.

• Find the value of A that make p.d.f
• What is the probability that the number of minutes that she will talk over the phone is (i) more than 10 minutes (ii) less than 5 minutes (iii) between 5 and 10 minutes?

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – STATISTICSFIRST SEMESTER – NOVEMBER 2012ST 1503/ST 1501 – PROBABILITY AND RANDOM VARIABLES
Date : 10/11/2012 Dept. No.         Max. : 100 Marks                 Time : 1:00 – 4:00                                               PART – AAnswer ALL the questions:                  (10 x 2 = 20 marks)1. Give the empirical definition of probability. Mention any one of its limitations.2. Define an event with reference to a random experiment. Give an example.3. Define conditional probability. When does this probability become Zero?   4. When do we say that an event A is statistically independent with respect to an event B? 5. Distinguish between pair wise independence and mutual independence.6. State multiplication law of probability for any two events. Hence, write down the law for      independent events.7. Find the E(X) of X whose pdf is f(x) =  ,  x >0.8. Write down the importance of Chebyshev’s inequality.9. Mention the advantages of generating functions.10. If X is random variable with mean 0 and variance 25, give an upper bound for the       probability P (│X – μ│ > 25 ).
PART – B
Answer any FIVE questions:          (5×8=40 marks)
11. Define Moment Generating Function (MGF). Show that MGF can be used to obtain the mean and        variance of a random variable.12. State and prove the addition theorem for two events. Hence, show that probability of a sure       event is One.   13. For any two events A and B, show that                                                                               P (AB C ) = P (A C) + P (B C) – P (AB C)14. If two dice are thrown, what is the probability that the ‘sum’ is (a) greater than 8 and (b) either 7 or 1115. If X is random variable with pdf f(x)=  , 2 ≤ x ≤4. Find the value of k.  Hence find Standard      Deviation of X.

 

16. Sixty per cent of the employees of the XYZ Corporation are college graduates. Of these, ten per cent       are in sales. Of the employees who did not graduate from the college, eighty per cent are in sales.     (i) What is the probability that an employee selected at random is in sales?     (ii) What is the probability that an employee selected at random is neither in sales nor a           college graduate?     17. Illustrate through an example that pair-wise independence does not imply mutual independence.18. The diameter of an electric cable is assumed to be a continuous random variable X with pdf f(x) = k        x(1-x) if  0  x  1 and 0, otherwise.  (i) Find the value of k. (ii) Obtain the distribution function of X.         (iii) Determine the value of constant c such that P[X< c] = P[X>c].
PART – C
Answer any TWO questions:          (2×20=40 marks)
19. (a) If two events A and B independent, show that their complements are independent of            each other.       (b) The probability of the wife who is 40 years old  living till 70 is   and the probability of              the husband who is now 50 living till 80 is  .  Find the probability that (a) only one will be alive for 30 years  (b) at least one will be alive for 30 years.
20. (a) State and prove Bayes theorem.       (b) Three identical urns contain the following proportion of balls. Urn1 : 2black, 1white.Urn2 : 1black, 2white.Urn3 : 2black, 2white.            An urn is selected at random and a ball is drawn. This ball turns out to be white. What is the            probability of drawing a white ball again if the first white ball drawn is not replaced.     21. (a) State and Prove Chebyshev’s Inequality.       (b) If X is a random variable with pdf f(x) =  ,  < x< , obtain an upper bound for the                probability P[│X – μ│ >  ]. Compare it with the actual probability.22. (a)  For any three events A. B and C, such B , and P(A) >0, show that  P (B A ) ≤ P (C A).(b) If A and B are two events such that P(A) =  and P(B) = , show that    (i) P (A )      and       (ii)     P (A )

 

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