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)
- What do you mean by “Random Experiments”?
- 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.
- State Classical definition of probability.
- 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.
- 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.
- Two coins are tossed. Show that the event “ head on first coin “ and event “ Coins fall alike”
are independent.
- 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.
- If for three mutually exclusive and exhaustive events A,B and C; and
P(B) = 2/3 P(C) then find P(A).
- Find the standard deviation of the probability distribution
|
x = x |
0 | 1 | 2 |
| P(x) |
- 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)
- 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).
- Prove that
- P( B ) = P(B) – P(A B)
- P(A ) = P(A) – (A B).
- In a random arrangement of the letters of the word “MATHEMATICS”, find the
probability that all the vowels come together.
- If events A and B are independent then prove that the complementary events and are
also independent. Also show that A and are independent.
- 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.
- State and prove multiplication law of probability when the events are (i ) not independent
(ii) independent.
- 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
- Determine the constant k
- Determine F(x).
- 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
- 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)
- 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?
- 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 ?
- 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.
- b) For n events A1, A2 … An , Prove that
(i) P (Ai) – (n-1)
(ii) P (Ai).
- a) State and prove Baye’s Theorem.
- 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’?
- 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?
- b) State and prove Chebyshev’s inequality?
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