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Unformatted text preview: Prof. Edwin Romeijn ESI 6321 Applied Probability Methods For Engineers Midterm Review Spring 2008 1/16 ESI 6321 APPLIED PROBABILITY METHODS FOR ENGINEERS Midterm review Spring 2008 Exercise 1 (Utility theory) Consider an urn that contains 99 white balls and 1 black ball. You are offered a chance to play a game in which a ball will be drawn from this urn. Each ball is equally likely to be drawn. If a white ball is drawn, you must pay $10,000. If the black ball is drawn, you will receive $1,000,000. Should you play the game? Let L 1 be the lottery where we dont play the game and L 2 be the lottery where we do. If the expected utility of L 1 is greater than the expected utility of L 2 , then we shouldnt play the game. With a probability of 1, we dont lose or win any money (ie, there is no change to our net worth). L 1 1 $0 With a probability of 0.99, well pick a white ball and our net worth will decrease by $10,000; with a probability of 0.01, well get that one black ball and our net worth will increase $1,000,000. 0.99 $-10,000 L 2 0.01 $1,000,000 E( U of L 1 ) = 1 ($0) = $ 0 E( U of L 2 ) = 0.99(-10,000) + 0.01(1,000,000) = -9,900 + 10,000 = $ 100 Since the E( U of L 2 ) > E( U of L 1 ), we would prefer L 2 over L 1 and should play the game. Prof. Edwin Romeijn ESI 6321 Applied Probability Methods For Engineers Midterm Review Spring 2008 2/16 Exercise 2 (Utility theory) You are a student at the University of Florida and desperately want to attend the next football game. Another student has a ticket and will sell it to you (at cost) for $10. However, you only have $5. You can bet the $5 in a poker game and thinking that youre a better-than-average player, you can win with probability 60%, doubling your money. Or, you could lose and end up with nothing. Based on the actual value of your money, would you be willing to play in the poker game? Let L 1 be the lottery where we dont play poker and L 2 be the lottery where we do. If the expected utility of L 1 is greater than the expected utility of L 2 , then we shouldnt play the game. If we dont play poker, with a probability of 1, well still have the $5 in our pocket (but we wont get to attend the football game). L 1 1 $5 If we do play poker, with a probability of 0.40, well lose and end up with nothing; with a probability of 0.60, well win and have $10. 0.40 $0 L 2 0.60 $10 E( U of L 1 ) = 1 ($5) = $5 E( U of L 2 ) = 0.40(0) + 0.60(10) = 0 + 6 = $6 Since the E( U of L 2 ) > E( U of L 1 ), we would prefer L 2 over L 1 and should play poker. Alternatively, we could look at the change in our money. If we dont play, with a certainty of 1, there will be no change to the money we have....
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