Problems 23610ch16ptg14 620 pm 16 22 chapter 16 markov

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sonal reasons, stays as a middle manager, stays as a senior manager. Problems 23610_ch16_ptg01_Web.indd 21 01/10/14 6:20 PM
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16-22 Chapter 16 Markov Processes a. What states are considered absorbing states? Why? b. Interpret the transition probabilities for the middle managers. c. Interpret the transition probabilities for the senior managers. d. What percentage of the current middle managers will eventually retire from the com- pany? What percentage will leave the company for personal reasons? e. The company currently has 920 managers: 640 middle managers and 280 senior man- agers. How many of these managers will eventually retire from the company? How many will leave the company for personal reasons? 16. Players in a particular sports league are classi fi ed based on the amount of time they play as backups or starters. Players can also become seriously injured or they can retire from competition. Consider the following transition probability matrix: Leaves— Middle Senior Retirement Personal Manager Manager Retirement 1.00 0.00 0.00 0.00 Leaves—Personal 0.00 1.00 0.00 0.00 Middle Manager 0.03 0.07 0.80 0.10 Senior Manager 0.08 0.01 0.03 0.88 Backup Starter Injured Retired Backup 0.4 0.4 0.1 0.1 Starter 0.1 0.5 0.15 0.25 Injured 0 0 1 0 Retired 0 0 0 1 a. What state(s) are absorbing states? b. What percentage of current starters will eventually be injured? c. Currently a team called The Sharks has eight backups and fi ve starters. How many of the players do you expect to end up injured? Retired? Case Problem 1 DEALER’S ABSORBING STATE PROBABILITIES IN   BLACKJACK The game of blackjack (sometimes called “21”) is a popular casino game. The goal is to have a hand with a value of 21 or as close to 21 as possible without exceeding 21. The player and the dealer are each dealt two cards initially. Both the player and dealer may draw additional cards (called “taking a hit”) in order to improve their hand. If either the player or dealer takes a hit and the value of the hand exceeds 21, the player or dealer is said to have gone broke and loses. Face cards and tens count 10 points, aces can be counted as 1 or 11, and all other cards count at their face value. The dealer’s advantage is that the player must decide on whether to take a hit fi rst. The player who takes a hit and goes over 21 goes broke and loses, even if the dealer later goes broke. For instance, if the player has 16 and draws any card with a value higher than a 5, the player goes broke and loses. For this reason, players will often decide not to take a hit when the value of their hand is 12 or greater. The dealer’s hand is dealt with one card up and one card down. So, the player’s deci- sion of whether to take a hit is based on knowledge of the dealer’s up card. A gambling professional asks you to help determine the probability of the ending value of the dealer’s 23610_ch16_ptg01_Web.indd 22 01/10/14 6:20 PM
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16-23 hand given different up cards. House rules at casinos require that the dealer continue to take a hit until the dealer’s hand reaches a value of 17 or higher. Having just studied Markov processes, you suggest that the dealer’s process of taking hits can be modeled as a Markov process with absorbing states.
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  • Spring '18
  • Markov process, Markov chain, Andrey Markov, Markov decision process

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