Concept explainers
First Paradox: Under certain circumstances, you have your best chance of winning a tennis
tournament if you play most of your games against the best possible opponent.
Alice and her two sisters, Betty and Carol, are avid tennis players. Betty is the best of the three sisters, and Carol plays at the same level as Alice. Alice defeats Carol 50% of the time but only defeats Betty 40% of the time.
Alice’s mother offers to give her $100 if she can win two consecutive games when playing three alternating games against her two sisters. Since the games will alternate, Alice has two possibilities for the sequence of opponents. One possibility is to play the first game against Betty, followed by a game with Carol, and then another game with Betty. We will refer to this sequence as BCB. The other possible sequence is CBC.
Make a guess of the best sequence for Alice to choose—-the one having the majority of the games against the weaker opponent or the one having the majority of the games against the stronger opponent.
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Finite Mathematics & Its Applications (12th Edition)
- Each Friday afternoon, Xiang and Lin both hand out coupons on campus for discount meals at a restaurant located in downtown Iowa City. The restaurant pays Xiang and Lin for their services: Depending on the number of coupons redeemed (used) by customers, Xiang and Lin possibly earn either nothing, $20 (this happens on 20% of Fridays for Xiang and on 40% of Fridays for Lin), or $40 (this happens on 40% of Fridays for Xiang and on 20% of Fridays for Lin.) Xiang works east of the river and Lin works west of the river so their earnings are independent. Find Xiang's mean earnings per Friday.arrow_forwardBob and Doug play a lot of Ping-Pong, but Doug is a much better player, and wins 60% of their games. To make up for this, if Doug wins a game he will spot Bob five points in their next game. If Doug wins again he will spot Bob ten points the next game, and if he still wins the next game he will spot him fifteen points, and continue to spot him fifteen points as long as he keeps winning. Whenever Bob wins a game he goes back to playing the next game with no advantage. It turns out that with a five-point advantage Bob wins 70% of the time; he wins 80% of the time with a ten-point advantage and 90% of the time with a fifteen-point advantage. Model this situation as a Markov chain using the number of consecutive games won by Doug as the states. There should be four states representing zero, one, two, and three or more consecutive game won by Doug. Find the transition matrix of this system, the steady-state vector for the system, and determine the proportion of games that Doug will win in…arrow_forwardBob and Doug play a lot of Ping-Pong, but Doug is a much better player, and wins 60% of their games. To make up for this, if Doug wins a game he will spot Bob five points in their next game. If Doug wins again he will spot Bob ten points the next game, and if he still wins the next game he will spot him fifteen points, and continue to spot him fifteen points as long as he keeps winning. Whenever Bob wins a game he goes back to playing the next game with no advantage. It turns out that with a five-point advantage Bob wins 40% of the time; he wins 70% of the time with a ten-point advantage and 70% of the time with a fifteen-point advantage. Model this situation as a Markov chain using the number of consecutive games won by Doug as the states. There should be four states representing zero, one, two, and three or more consecutive games won by Doug. Find the transition matrix of this system, the steady-state vector for the system, and determine the proportion of games that Doug will win in…arrow_forward
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