Let A1, A2,...,An be a sequence of increasing events of a sample space. That is, Ai is a subset of Ai+1 i 1. Let B1=A1 and Bi = Ai - Ai-1. Note that Ui=1 to Ai = Ui=1 to Bi
a. Use the events B1, B2, ... to prove that limn P(An) = P(limn An)
b. Suppose an experiment consists of selecting a random point from the interval (1,2). Use past (a) and Demorgan's Laws to find the probability of choosing 1.657555... from the interval (1,2).
c. In a series of games, the winning number of the nth game n = 1, 2, 3... is a number selected at random from the set of integers (1, 2, ..., n+2). Jim bets on 1 in each game and says he will quit as soon as he wins. Use part (a) to find the probability that he has to play indefinitely.
The limits of a sequence of events to obtain is determined.