Concept explainers
Consider the following card game. The player and dealer each receive a card from a 52-card deck. At the end of the game the player with the highest card wins; a tie goes to the dealer. (You can assume that Aces count 1, Jacks 11, Queens 12, and Kings 13.) After the player receives his card, he keeps the card if it is 7 or higher. If the player does not keep the card, the player and dealer swap cards. Then the dealer keeps his current card (which might be the player’s original card) if it is 9 or higher. If the dealer does not keep his card, he draws another card. Use simulation with at least 1000 iterations to estimate the probability that the player wins. (Hint: See the file Sampling Without Replacement.xlsx, one of the example files, to see a clever way of simulating cards from a deck so that the Same card is never dealt more than once.)
Want to see the full answer?
Check out a sample textbook solutionChapter 11 Solutions
Practical Management Science
- In this version of dice blackjack, you toss a single die repeatedly and add up the sum of your dice tosses. Your goal is to come as close as possible to a total of 7 without going over. You may stop at any time. If your total is 8 or more, you lose. If your total is 7 or less, the house then tosses the die repeatedly. The house stops as soon as its total is 4 or more. If the house totals 8 or more, you win. Otherwise, the higher total wins. If there is a tie, the house wins. Consider the following strategies: Keep tossing until your total is 3 or more. Keep tossing until your total is 4 or more. Keep tossing until your total is 5 or more. Keep tossing until your total is 6 or more. Keep tossing until your total is 7 or more. For example, suppose you keep tossing until your total is 4 or more. Here are some examples of how the game might go: You toss a 2 and then a 3 and stop for total of 5. The house tosses a 3 and then a 2. You lose because a tie goes to the house. You toss a 3 and then a 6. You lose. You toss a 6 and stop. The house tosses a 3 and then a 2. You win. You toss a 3 and then a 4 for total of 7. The house tosses a 3 and then a 5. You win. Note that only 4 tosses need to be generated for the house, but more tosses might need to be generated for you, depending on your strategy. Develop a simulation and run it for at least 1000 iterations for each of the strategies listed previously. For each strategy, what are the two values so that you are 95% sure that your probability of winning is between these two values? Which of the five strategies appears to be best?arrow_forwardYou now have 5000. You will toss a fair coin four times. Before each toss you can bet any amount of your money (including none) on the outcome of the toss. If heads comes up, you win the amount you bet. If tails comes up, you lose the amount you bet. Your goal is to reach 15,000. It turns out that you can maximize your chance of reaching 15,000 by betting either the money you have on hand or 15,000 minus the money you have on hand, whichever is smaller. Use simulation to estimate the probability that you will reach your goal with this betting strategy.arrow_forwardYou have 5 and your opponent has 10. You flip a fair coin and if heads comes up, your opponent pays you 1. If tails comes up, you pay your opponent 1. The game is finished when one player has all the money or after 100 tosses, whichever comes first. Use simulation to estimate the probability that you end up with all the money and the probability that neither of you goes broke in 100 tosses.arrow_forward
- Suppose that you have two four-sided dice that are each equally weighted (i.e. equal chance of any of the four sides landing face up). One of them has values [12.5968, 2, -2 and -12.5968] , the other [12.5968, 4, -2 and -12.5968] . Which of them has a µ of zero using the probability model approach. 1. [12.5968, 4, -2 and -12.5968] 2. [12.5968, 2, -2 and -12.5968]arrow_forwardPlayer A and B play a game in which each has three coins, a 5p, 10p and a 20p. Each selects a coin without the knowledge of the other’s choice. If the sum of the coins is an odd amount, then A wins B’s coin. But, if the sum is even, then B wins A’s coin. Find the best strategy for each player and the values of the game.arrow_forwardA gambler in Las Vegas is cutting a deck of cards for $1,000. What is the probability that the card for the gambler will be the following? a. A face card b. A queen c. A spade d. A jack of spadesarrow_forward
- Consider a population of 3000 people, 1400 of whom are men. Assume that 700 of the women in this population earn at least $60,000 per year, and 500 of the men earn less than $60,000 per year. a. What is the probability that a randomly selected person from this population earns less than $60,000 per year? b. If a randomly selected person is observed to earn less than $60,000 per year, what is the probability that this person is a man? c. If a randomly selected person is observed to earn at least $60,000 per year, what is the probability that this person is a woman?arrow_forwardSuppose that two firms, an incumbent and a potential entrant, compete in a market. If they both operate, the resulting profits for the incumbent and the potential entrant are $30 million and $5 million, respectively. If the potential entrant decides not to enter the market, the resulting profits for the incumbent and potential entrant are $100 million and $0, respectively. Assume that it is not an option for the incumbent to exit the market. 1) Which of the two possible outcomes (for the potential entrant to enter or not enter) are Pareto efficient? 2) Now suppose there are side payments available. Which of the two possible outcomes is Pareto efficient? 3) Is it possible that the incumbent will agree to pay for the introduction of side payments? 4) Now suppose that side payments are allowed, but it costs $80 million in legal fees to actually use them. Which of the two possible outcomes (for the potential entrant to enter or not enter) are Pareto efficient allocations? (For simplicity,…arrow_forwardConsider a cheap talk game in which Nature chooses the sender’s type andthere are three feasible types: x, y, and z, which occur with probability 1/4, 1/4,and 1/2, respectively. The sender learns her type and then chooses one of fourpossible messages: m1, m2, m3, or m4. The receiver observes the sender’smessage and chooses one of three actions: a, b, or c. The payoffs are shown in the table below. a. Suppose the sender’s strategy is as follows: (1) if the type is x, thenchoose message m1, and (2) if the type is y or z, then choose messagem2. The receiver’s strategy is the following: (1) if the message is m1,then choose action a; (2) if the message is m2, then choose action b;and (3) if the message is m3 or m4, then choose action a. For appropriatelyspecified beliefs for the receiver, show that this strategy pair is aperfect Bayes–Nash equilibrium.b. Find a separating perfect Bayes–Nash equilibrium.arrow_forward
- You now have 10,000, all of which is invested in a sports team. Each year there is a 60% chance that the value of the team will increase by 60% and a 40% chance that the value of the team will decrease by 60%. Estimate the mean and median value of your investment after 50 years. Explain the large difference between the estimated mean and median.arrow_forwardA martingale betting strategy works as follows. You begin with a certain amount of money and repeatedly play a game in which you have a 40% chance of winning any bet. In the first game, you bet 1. From then on, every time you win a bet, you bet 1 the next time. Each time you lose, you double your previous bet. Currently you have 63. Assuming you have unlimited credit, so that you can bet more money than you have, use simulation to estimate the profit or loss you will have after playing the game 50 times.arrow_forwardA winner of the multi - state lotto won a one-time payout and decided to invest part of it into an annuity. If the winner invests $6, 700,000.00 into a 30 year annuity that pays 3.5%, compounded monthly and makes each month payments. What is the amount of each month payments? The payment would be $ . (Round to 2 decimal places.)arrow_forward
- Practical Management ScienceOperations ManagementISBN:9781337406659Author:WINSTON, Wayne L.Publisher:Cengage,