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 )**

- Distinguish between Mutually Exclusive event and Independent event.
- Prove that for any event A in S, P( A ∩ A
^{C}) = 0. - 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.
- A fair coin is tossed 5 times. What is the probability of having at least one head?
- Prove that if P(A) > P(B), then P(A│B) > P(B│A)
- If two events A and B are independent, show that (i) A
^{C}and B^{C}are independent. - Out of 800 families with 4 children each, how many families would be expected to have at least one boy?
- 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.
- A fair coin is tossed three times. Let X be the number of tails appearing. Find the probability distribution of X . Calculate E(X).
- State addition theorem of probability for two events A and B.

__PART –B__

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

- 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.

- 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.
- 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.

- 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.

- 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?

- 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

- a) State and prove Baye’s Theorem.
- b) If A, B, C are mutually independent events then prove that and C are also independent.
- 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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