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.
How would you explain to someone who didn’t know probability why the sequence that you chose is best?
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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
- Bob and Doug play a lot of Ping-Pong, but Doug is a much better player, and wins 80% 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 20% of the time; he wins 50% of the time with a ten-point advantage and 80% 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_forwardBob and Doug play a lot of Ping-Pong, but Doug is a much better player, and wins 80% 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 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 games won by Doug. Find the transition matrix of this system, the steady-state vector for system, and determine the proportion games that Doug will win the long…arrow_forwardSuppose you decided to play a gambling game. In order to play the game there is a $1.50 dollar fee to play. If you roll a 1, 2, or 3 you win nothing (i.e., your net profit is $-1.50). If you roll a 4 or 5, you win $3.50 (i.e., your net profit is $2.00). If you roll a 6 you win $5.00 (i.e., your net profit is $3.50).Use the information described above to construct a probability distribution table for the random variable xx which represents the net profit of your winnings. Note: Be sure to enter your probabilities as reduced fractions. Die Roll xx P(x) Roll a 1, 2, or 3 Roll a 4 or 5 Roll a 6 Find the amount you would expect to win or lose each time you played the game. Round your final answer to two decimal places.μ=arrow_forward
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- Problem. Two players A and B play a fair game such that the player who wins a total of 6 rounds first wins a prize. Suppose the game unexpectedly stops when A has won a total of 5 rounds and B has won a total of 3 rounds. How should the prize be divided between A and B?arrow_forwardSuppose you decided to play a gambling game. In order to play the game there is a $1.50 dollar fee to play. If you roll a 1, 2, or 3 you win nothing (i.e., your net profit is $-1.5 dollars). If you roll a 4 or 5, you win $2.50 (i.e., your net profit is $1). If you roll a 6 you win $5.75 (i.e., your net profit is $4.25).a) Use the information described above to constuct a probability distribution table for the random variable xx which represents the net profit of your winnings. Note: Be sure to enter your probabilities as reduced fractions. xx P(x)P(x) (You roll a 1,2,or 3) (You roll a 1,2, or 3) (You roll a 4 or 5) (You roll a 4 or 5) (You roll a 6) (You roll a 6) b) Find the amount you would expect to win or lose each time you played the game. Round your final answer to two decimal places.μ=arrow_forwardYou are playing in a game show, and have won $1,200 so far. Now it's the final round, and you must choose to play double-or-nothing game. The host offers you to play the "Monty Hall" problem with 4 boxes, where only one box contains the "double" prize. You should choose a box, then the host opens two empty boxes and gives you the chance to switch boxes or to stick to your first choice. Assuming you choose the best strategy, what are your chances of doubling your prize? (You should enter a value is in the range 0-1)arrow_forward
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,