Chapter 1 Homework Solutions

Ch. 1.4 Problem 10

Prove:

Solution:

Given P(AÈB) £ 1, we have by Theorem 1.6,

Solving this for P(AB) completes the proof.

Problem 20

The coefficients of the quadratic equation are determined by tossing a fair die twice (the first outcome is b, the second outcome is c). Find the probability that the equation has real roots.

Solution: From the quadratic formula, we know that the roots are real iff . This problem is solved using brute force. For the 36 ordered pairs (b, c) that are the possible outcomes, we find 19 outcomes (shaded below) satisfy the inequality. Therefore the probability is .

Ch. 1.7 Problem 12 The Borel-Cantelli Lemma

Let {A₁, A₂, A₃, ...} be a sequence of events. Prove that if the series

converges, then

Solution: First we need to understand the problem, and in this case that means try to get some idea of what the expression

means. This is obviously the probability of a set constructed as indicated inside of the parenthesis. (To do this you might try writing out an example. Try the case where P(A_n) = .) Now consider the expression

This is written out as:

To simplify let,

, and

By Boole’s inequality, (see Ch 1.4 problem 28) . Since converges, we have:

(This last equation states the fact that remainder term of a convergent infinite series approaches zero as n goes to infinity). By construction,

so we have by Theorem 1.8

Q.E.D.