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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