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MSO 201A / ESO 209: Probability and Statistics

2012-2013-II Semester

DIPA-I

(Discussion of Illustrative Problems and Assignment-I)

Instructor: Neeraj Misra

Note: (i) Abbreviation DP (or AP) against each problem stands for the Discussion

Problem (or the Assignment Problem).

(ii) To the extent possible, Discussion Problems will be solved by tutors during the tutorial

hours. Assignment problems are to be considered as Practice Problems for students.

Students may discuss these problems with tutors during office hours.

1. (AP) Let Ω = {1, 2, 3, 4}. Check which of the following is a sigma-field of subsets

of Ω:(a) F 1 = {φ, {1, 2}, {3, 4}}; (b) F 2 = {φ, Ω, {1}, {2, 3, 4}, {1, 2}, {3, 4}};

(c) F 3 = {φ, Ω, {1}, {2}, {1, 2}, {3, 4}, {2, 3, 4}, {1, 3, 4}}.

2. (AP) Show that a class A of subsets of Ω is a sigma-field of subsets of Ω if, and

only if, the following three conditions are satisfied: (i) Ω ∈ A; (ii) A ∈ A ⇒ Ac =

Ω − A ∈ A; (iii) An ∈ A, n = 1, 2, . . . ⇒ ∩∞

n=1An ∈ A.

3. (AP) Let {F λ; λ ∈ Λ} be a collection of sigma-fields of subsets of Ω.

(a) Show that

λ∈Λ F λ is a sigma-field;(b) Using a counter example show that

λ∈Λ F λ may not be a sigma-field.

4. (AP) Let Ω be an infinite set and let A = {A ⊆ Ω : A is finite or Ac is finite}.

(a) Show that A is closed under complements and finite unions;

(b) Using a counter example show that A may not be closed under countably infinite

unions (and hence A may not be a sigma-field).

5. (DP) Let Ω be an uncountable set and let A = {A ⊆ Ω : A is countable or Ac is countable}.

(a) Show that A is a sigma-field;

(b) What can you say about A when Ω is countable?

6. (DP) Let F = P (Ω) = power set of Ω = {0, 1, 2, . . .}. In each of the following cases,

verify if (Ω, F , P ) is a probability space:

(a) P (A) =

x∈A e−λλx/x!, A ∈ F , λ > 0;

(b) P (A) =

x∈A p(1 − p)x, A ∈ F , 0 < p


2/17

7. (AP) Let (Ω, F , P ) be a probability space and let A, B,C,D ∈ F . Suppose that

P (A) = 0.6, P (B) = 0.5, P (C ) = 0.4, P (A ∩ B) = 0.3, P (A ∩ C ) = 0.2, P (B ∩ C ) =

0.2, P (A ∩ B ∩ C ) = 0.1, P (B ∩ D) = P (C ∩ D) = 0, P (A ∩ D) = 0.1 and P(D)=0.2.

Find:(a) P (A ∪ B ∪ C ) and P (Ac ∩ Bc ∩ C c); (b) P ((A ∪ B) ∩ C ) and P (A ∪ (B ∩ C ));

(c) P ((Ac ∪ Bc) ∩ C c) and P ((Ac ∩ Bc) ∪ C c); (d) P (D ∩ B ∩ C ) and P (A ∩ C ∩ D);

(e) P (A ∪ B ∪ D) and P (A ∪ B ∪ C ∪ D); (f) P ((A ∩ B) ∪ (C ∩ D)).

8. (AP) Let (Ω, F , P ) be a probability space and let A and B be two events (i.e.,

A, B ∈ F ).

(a) Show that the probability that exactly one of the events A or B will occur is

given by P (A) + P (B) − 2P (A ∩ B);

(b) Show that P (A ∩ B) − P (A)P (B) = P (A)P (Bc) − P (A ∩ Bc) = P (Ac)P (B) −

P (Ac ∩ B) = P ((A ∪ B)c) − P (Ac)P (Bc).

9. (AP) Suppose that n (≥ 3) persons P 1, . . . , P n are made to stand in a row at

random. Find the probability that there are exactly r persons between P 1 and P 2;

here r ∈ {1, 2, . . . , n − 2}.

10. (DP) A point (X, Y ) is randomly chosen on the unit square S = {(x, y) : 0 ≤

x ≤ 1, 0 ≤ y ≤ 1} (i.e., for any region R ⊆ S for which the area is defined,

the probability that (X, Y ) lies on R is area of Rarea of S ). Find the probability that the

distance from (X, Y ) to the nearest side does not exceed 13 units.

11. (AP) Three numbers a, b and c are chosen at random and with replacement from the

set {1, 2, . . . , 6}. Find the probability that the quadratic equation ax2 + bx + c = 0

will have real root(s).

12. (DP) Three numbers are chosen at random from the set {1, 2, . . . , 50}. Find the

probability that the chosen numbers are in

(a) arithmetic progression;

(b) geometric progression.

13. (AP) Consider an empty box in which four balls are to be placed (one-by-one)according to the following scheme. A fair die is cast each time and the number of

spots on the upper face is noted. If the upper face shows up 2 or 5 spots then a

white ball is placed in the box. Otherwise a black ball is placed in the box. Given

that the first ball placed in the box was white find the probability that the box will

contain exactly two black balls.

14. (DP) Let ((0, 1], F , P ) be a probability space such that F contains all subintervals

of (0, 1] and P ((a, b]) = b − a, where 0 ≤ a < b ≤ 1.

2


3/17


4/17

20. (AP) Let A, B and C be three events such that A and B are negatively (positively)

associated and B and C are negatively (positively) associated. Can we conclude

that, in general, A and C are negatively (positively) associated?

21. (AP) Let (Ω, F , P ) be a probability space and let A and B be two events (i.e.,

A, B ∈ F ). Show that if A and B are positively (negatively) associated then A and

Bc are negatively (positively) associated.

22. (DP) A locality has n houses numbered 1, . . . , n and a terrorist is hiding in one

of these houses. Let H j denote the event that the terrorist is hiding in house

numbered j, j = 1, . . . , n, and let P (H j) = p j ∈ (0, 1), j = 1, . . . , n. During a

search operation, let F j denote the event that search of the house number j will fail

to nab the terrorist there and let P (F j|H j) = r j ∈ (0, 1), j = 1, . . . , n. For each

i, j ∈ {1, . . . , n}, i = j, show that H j and F j are negatively associated but H i and

F j are positively associated. Interpret these findings.

23. (AP) Let A, B and C be three events such that P (B ∩ C ) > 0. Prove or disprove

each of the following:

(a) P (A ∩ B|C ) = P (A|B ∩ C )P (B|C ); (b) P (A ∩ B|C ) = P (A|C )P (B|C ) if A and

B are independent events.

24. (DP) A k-out-of-n system is a system comprising of n components that functions if

and only if at least k (k ∈ {1, 2, . . . , n}) of the components function. A 1-out-of-n

system is called a parallel system and an n-out-of-n system is called a series system .Consider n components C 1, . . . , C n that function independently. At any given time

t the probability that the component C i will be functioning is pi(t) (∈ (0, 1)) and

the probability that it will not be functioning at time t is 1 − pi(t), i = 1, . . . , n.

(a) Find the probability that a parallel system comprising of components C 1, . . . , C n

will function at time t;

(b) Find the probability that a series system comprising of components C 1, . . . , C n

will function at time t;

(c) If pi(t) = p(t), i = 1, . . . , n find the probability that a k-out-of-n system com-

prising of components C 1, . . . , C n will function at time t.

4


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