Problem Sheet 0
Statistics
2008
Department of Mathematics, IIT Madras
1. Show that P (A) + P (B) − 1 ≤ P (A ∩ B) ≤ P (A) + P (B).
2. An experiment has five possible outcomes: A, B, C, D, E. Check whether the following
assignment of probabilities are permissible:
(a) P (A) = 0.2, P (B) = 0.2, P (C) = 0.2, P (D) = 0.2, P (E) = 0.2.
(b) P (A) = 0.21, P (B) = 0.26, P (C) = 0.58, P (D) = 0.01, P (E) = 0.06.
(c) P (A) = 0.18, P (B) = 0.19, P (C) = 0.2, P (D) = 0.21, P (E) = 0.22.
(d) P (A) = 0.1, P (B) = 0.3, P (C) = 0.1, P (D) = 0.6, P (E) = −0.1.
(e) P (A) = 0.23, P (B) = 0.12, P (C) = 0.05, P (D) = 0.5, P (E) = 0.08.
3. A hat contains 20 white slips of paper numbered from 1 through 20, ten red slips of paper
numbered 1 through 10, 40 yellow slips of paper numbered 1 through 40, and 10 blue
slips of paper numbered 1 through 10. If these 80 slips of paper are suffled throughly so
that each sl
17.ip has the same probability of being drawn, find the probabilities of
drawing a slip of paper that is
(a) blue or white;
(b) numbered 1, 2, 3, 4, or 5;
(c) red or yellow and numbered 1, 2, 3, or 4;
(d) numbered 5, 15, 25, or 35;
(e) white and numbered higher than 12, or yellow and numbered higher than 26.
4. At Rounak college it is known that 1/3 of the students live off campus, 5/9 of the students
are from A.P., and 3/4 of the students are from outside of A.P. or live on campus. What
is the probability that a student selected at random from Rounak college is from out of
A.P. and lives on campus?
5. Show by means of numerical examples that P (B|A) + P (B|A0 ) (a) may be equal to 1; (b)
need not be equal to 1.
6. Suppose P (B|A) = P (B) 6= 0. What can you say about P (A|B)?
7. Give a numerical example to show that P (A ∩ B ∩ C) = P (A)P (B)P (C) does not necessarily imply that the events A, B, C are independent.
8. Show that if A, B, C are independent events, then so are A, B ∩ C, and so are A, B ∪ C.
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9. With reference to Problem 4, what is the probability that one of the students will be
living on campus given that he is from out of A.P.?
10. It is felt that the probabilities are 0.2, 0.4, 0.3, 0.1 that the basket ball teams of four
IITs B,C,D,G will win their championship. If IIT C is placed on probation and declared
inelligible for championship, what is the probability that IIT B will win the championship?
11. The probability of surviving a certain transplant operation is 0.55. If a patient survives
the operation, the probability that his body will reject the transplant within a month is
0.2. What is the probability of surviving both of these critical stages?
12. Suppose that in Kashmir, the probability that a rainy fall day is followed by a rainy day
is 0.8 and the probability that a sunny fall day is followed by a rainy day is 0.6. Find the
probabilities that a rainy fall day is followed by
(a) a rainy day, a sunny day, and another rainy day;
(b) two sunny days and then a rainy day;
(c) two rainy days and then two rainy days;
(d) rain two days later.
13. A coin is loaded so that the probabilities of heads and tails are 0.52 and 0.48, rspectively.
If the coin is tossed three times, what are the probabilities of getting
(a) all heads; (b) two tails and a head in that order?
14. Medial record show that one out of 10 persons in a certain town has a thyroid deficieny.
If 12 persons in this town are randomly chosen and tested, what is the probability that
at least one of them will have a thyroid deficiency?
15. If a person randomly picks four of the 15 gold coins a dealer has in stock, and six of the
coins are counterfiets, what is the probability that the coins picked will all be counterfiets?
16. An art dealer receives a shipment of five old paintings from abroad, and on the basis of past
experience, she feels that the probabilities are, respectively, 0.76, 0.09, 0.02, 0.01, 0, 02, 0.10
that 0, 1, 2, 3, 4, 5 of them are forgeries. Since the cost of authentication is fairly high, she
decided to select one of the five paintings at random and send it away for authentication.
If it turns out that this painting is a forgery, what probability should she now assign to
the possibility that all other paintings are also forgeries?
Answers: 2. (a) yes (b) no (c) yes (d) no (e) no. Explain.
3. (a) 3/8 (b) 1/4 (c) 1/10 (d) 1/10 (e) 11/40
10. 1/3
11. 0.44
13. (a) 0.1406 (b) 0.1198
4. 13/36
6. P (A)
12. (a) 0.096 (b) 0.048 (c) 0.0512 (d) 0.76
14. 0.7176
15. 1/91
2
16. 0.6757
9. 13/16