Hwk-2 - i has a winning probability of p i , then if it...

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Homework 2 Due: Sep 18th 2008 1. Consider the gambler’s problem discussed in class. The gambler can bet N times, but now consider the case that the winning probabilities for the N bets are p 1 ,p 2 ,...,p N . That is, the winning probability of the first bet is p 1 , the second one is p 2 etc. Find the optimal gambling strategy in this case. The following two problems are variations of the first, and they are optional. 2. Consider the first problem, but assume that the winning probabilities can change. Specifically, let us assume that for the first bet, the probability of winning is p 1 . If the first one turns out to be a win, then the second bet only has winning probability of p 2 ( p 2 < p 1 ), and if it is a lose, then the next winning probability will remain p 1 . In general, for bet i , if bet
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Unformatted text preview: i has a winning probability of p i , then if it turns out to be a win, then the next bet will have winning probability of p 2 and if it is lose, the next one will have a winning probability of p 1 . Formulate this problem as a DP and try to solve it. 3. Suppose that, you do not have the winning probability in each play, but only know that, for the rst play, the winning probability is either p 1 or p 2 , with respective probabilities and 1- . What is your optimal gambling strategy in this case (hint: you need to periodically update the winning probability each and every time afteryou get). 4. Find the distance of the shortest path from Node 1 to Node 7 in the following graph using Dijkstras algorithm. 1...
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