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EE3110/EE5110 Problem Set-1

1. For two events A  and  B, let P( A) ≥  1 − δ and  P(B) ≥  1 − δ. Derive a lower bound

on P( AB), and hence show that if  A  and B  are events with probability nearly one,

so is AB.

2.   A,  B  and  C  are three events such that (i)  P( A)   >   P(B)   >   P(C)   >   0, (ii)  A  and

B  partition the sample space  Ω, and (iii)   A  and  C  are independent. Determine

whether  B  and C can be disjoint.

3. If three events A, B, and C are independent, show that the events A  and B ∪ C are

independent.

4. A die is rolled three times. Let Aij  be the event that the ith and  jth rolls produce

the same number, where 1  ≤   i  <   j  ≤  3. Determine whether the events { Aij} areindependent. Are they pairwise independent?

5. In a box, there are four red balls, six red cubes, six blue balls and an unknown

number of blue cubes. When an object from the box is selected at random, the

shape and colour of the object are independent. Determine the number of blue

cubes.

6. You are given that at least one of the events Ai, 1  ≤  i  ≤  n, is certain to occur, but

certainly no more than two can (jointly) occur. If  P( A

i) =   p

 and P( A

i A

 j) =   q

,i =  j, derive a lower bound on  p  and an upper bound on q (both) in terms of  n.

7. Given events  A,  B,  C, and  D, determine an expression for  P(“only A”) in terms

of  P( A)  and any joint probability involving these events, but  not   involving any

complements.

8. The event A is said to be attracted by the event  B  if   P( A|B) >  P( A) and repelled

 by B if   P( A|B) < P( A).

(a) Show that if  B  attracts A, then A  attracts B  and Bc repels A.

(b) If   A  attracts  B  and  B  attracts  C, show that   A  does not necessarily attract  C.

Come up with a example to substantiate this claim.

9.   A, B, and C  are three events. Show that   P( A|B) =  P( A|BC)α + P( A|BCc)(1− α),

where  α  is a conditional probability. Determine α  in terms of  A, B, and C.

10. Let  P( A) =   0.4,   P(B) =   0.3,   P(C) =   0.7,   P( AcB) =   0.1, and   P( ABCc) =   0.1.

Determine (a) P( ABc| A)   (b) P( AB)   (c) P(Bc ∪ C| A).

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11. There are N  urns of which the  rth contains  r − 1 red balls and  N  − r  blue balls.

You pick an urn at random and remove two balls at random without replacement.

Find the probability that

(a) the second ball is blue;

(b) the second ball is blue, given that the first is blue.

12. Given two σ -fields F 1 and F 2, show that their intersection is also a  σ -field.

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