MidtermReviewSolutions

MidtermReviewSolutions - Prof Edwin Romeijn ESI 6321...

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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 don’t 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 shouldn’t play the game. With a probability of 1, we don’t lose or win any money (ie, there is no change to our net worth). L 1 1 $0 With a probability of 0.99, we’ll pick a white ball and our net worth will decrease by $10,000; with a probability of 0.01, we’ll 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 you’re 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 don’t 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 shouldn’t play the game. If we don’t play poker, with a probability of 1, we’ll still have the $5 in our pocket (but we won’t get to attend the football game). L 1 1 $5 If we do play poker, with a probability of 0.40, we’ll lose and end up with nothing; with a probability of 0.60, we’ll 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 don’t play, with a certainty of 1, there will be no change to the money we have....
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This note was uploaded on 05/12/2010 for the course ESI 6321 taught by Professor Josephgeunes during the Spring '07 term at University of Florida.

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MidtermReviewSolutions - Prof Edwin Romeijn ESI 6321...

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