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.)
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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 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_forward
- You have made it to the final round of “Let’s Make aDeal.” You know there is $1 million behind either door 1,door 2, or door 3. It is equally likely that the prize is behindany of the three. The two doors without a prize have nothingbehind them. You randomly choose door 2, but before door2 is opened Monte reveals that there is no prize behinddoor 3. You now have the opportunity to switch and choosedoor 1. Should you switch? Assume that Monte plays asfollows: Monte knows where the prize is and will open anempty door, but he cannot open door 2. If the prize is reallybehind door 2, Monte is equally likely to open door 1 ordoor 3. If the prize is really behind door 1, Monte mustopen door 3. If the prize is really behind door 3, Montemust open door 1. What is your decision?arrow_forwardConsider the following Bargaining game, where each player tries to maximize his own number of coins. Player A starts the game with 3 coins and Player B starts with 1 coin. Player A needs to decide how many coins to give to Player B. Player A moves first, and makes an offer to Player B. Player B observes the offer, and then chooses to accept or reject the offer. If player B rejects the offer, the game is over: Player A loses all his 3 coins and player B losses his 1 coin (both players end up with zero coins). If Player B accepts the offer, the game is over: Player B keeps his original 1 coin plus the coin(s) offered by Player A, while Player A keeps the coin(s) he did not offer to player B. In the subgame perfect equilibrium of this game: Select the right option from below Player A offers zero coins to Player B Player A offers one coin to Player B Player A offers two coins to Player B Player A offers three coins to Player B Player A plays a mixed…arrow_forwardPlayer A begins by placing a checker on the lower left-hand corner of a checkerboard (8 by 8 squares). Player B places a checker one square to the right or one square up or one square diagonally up and to the right of Player A’s checker. Then A places a checker one square to the right or one square up or one square diagonally up and to the right of Player B. The players continue alternating moves in this way. The winner is the player who places a checker in the upper right corner. Would you rather be Player A or Player B? a) For this week’s homework, you are to find a winning strategy and express that winning strategy in a way that is easy to understand. b) Now generalize the board to any sized rectangle (mxn). What is your winning strategy now?arrow_forward
- A study on ethics in the workplace by the Ethics Resource Center and Kronos, Inc., revealed that 35% of employees admit to keeping quiet when they see coworker misconduct. Suppose 70% of employees who admit to keeping quiet when they see coworker misconduct call in sick when they are well. In addition, suppose that 45% of the employees who call in sick when they are well admit to keeping quiet when they see coworker misconduct. If an employee is randomly selected, determine the following probabilities:a. The employee calls in sick when well and admits to keeping quiet when seeing coworker misconduct.b. The employee admits to keeping quiet when seeing coworker misconduct or calls in sick when well.c. Given that the employee calls in sick when well, he or she does not keep quiet when seeing coworker misconduct.d. The employee neither keeps quiet when seeing coworker misconduct nor calls in sick when well.e. The employee admits to keeping quiet when seeing coworker misconduct and does not…arrow_forwardEmployees in the textile industry can be segmented as follows:Employees NumberFemale and union 12,000Female and nonunion 25,000Male and union 21,000Male and nonunion 42,000 What is the probability that an employee is male?arrow_forwardBased on Kelly (1956). You currently have 100. Each week you can invest any amount of money you currently have in a risky investment. With probability 0.4, the amount you invest is tripled (e.g., if you invest 100, you increase your asset position by 300), and, with probability 0.6, the amount you invest is lost. Consider the following investment strategies: Each week, invest 10% of your money. Each week, invest 30% of your money. Each week, invest 50% of your money. Use @RISK to simulate 100 weeks of each strategy 1000 times. Which strategy appears to be best in terms of the maximum growth rate? (In general, if you can multiply your investment by M with probability p and lose your investment with probability q = 1 p, you should invest a fraction [p(M 1) q]/(M 1) of your money each week. This strategy maximizes the expected growth rate of your fortune and is known as the Kelly criterion.) (Hint: If an initial wealth of I dollars grows to F dollars in 100 weeks, the weekly growth rate, labeled r, satisfies F = (I + r)100, so that r = (F/I)1/100 1.)arrow_forward
- You 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_forwardThe game of Chuck-a-Luck is played as follows: You pick a number between 1 and 6 and toss three dice. If your number does not appear, you lose 1. If your number appears x times, you win x. On the average, use simulation to find the average amount of money you will win or lose on each play of the game.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_forward
- Practical Management ScienceOperations ManagementISBN:9781337406659Author:WINSTON, Wayne L.Publisher:Cengage,