Page 74 - Probability and Statistical Inference

1. Notions of Probability 51

come up on the two dice. Also, find the probability of observing the red die
scoring higher than that from the yellow die.
1.1.3 Five equally qualified individuals consisting of four men and one woman
apply for two identical positions in a company. The two positions are filled by
selecting two from this applicant pool at random.
(i) Write down the sample space S for this random experiment;
(ii) Assign probabilities to the simple events in the sample space S;
(iii) Find the probability that the woman applicant is selected for a posi-
tion.
1.2.1 One has three fair dice which are red, yellow and brown. The three
dice are rolled on a table at the same time. Consider the following two events:
A : The sum total of the scores from all three dice is 10
B : The sum total of the scores from the red and brown dice exceeds 8
Find P(A), P(B) and P(A n B).
1.2.2 Prove the set relations given in (1.2.2).
1.2.3 Suppose that A, B, C are subsets of S. Show that
(i) A ∆ C ⊆ (A ∆ B) ∪ (B ∆ C);
(ii) (A ∆ B) ∪ (B ∆ C)
c
= [(A ∪ B) ∩ (A ∩ B) ] ∪ [(B ∪ C) ∩ (B ∩ C) ].
c
{Hint: Use the Venn Diagram.}
1.2.4 Suppose that A , ..., A are Borel sets, that is they belong to ß. Define
1
n
the following sets: B = A ,B = A ∩ B = A ∩ (A ∪ A ) , ..., B = A ∩ (A 1
c
2
2
1
1
3
n
1
n
3
2
c
∪ ... ∪ A ) . Show that
n–1
(i) B , ..., B are Borel sets;
1 n
(ii) B , ..., B are disjoint sets;
1 n
(iii)
1.2.5 Suppose that S = {(x, y) : x + y ≤ 1}. Extend the ideas from the
2
2
Example 1.2.3 to obtain a partition of the circular disc, S. {Hint: How about
considering A = {(x, y) : 1/2 < x + y = 1/2i–1}, i = 1, 2, ...? Any other
i
2
2
i
possibilities?}
1.3.1 Show that the Borel sigma-field ß is closed under the operation of (i)
finite intersection, (ii) countably infinite intersection, of its members. {Hint:
Can DeMorgan’s Law (Theorem 1.2.1) be used here?}
1.3.2 Suppose that A and B are two arbitrary Borel sets. Then, show that
A ∆ B is also a Borel set.