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Case Problem Dealer’s Absorbing State Probabilities in Blackjack The game of blackjack (sometimes called “21) is a popular casino game. The goal js 1, have a hand with a value of 21 or as close to 21 as possible without exceeding 21, Tp, player and the dealer are each dealt two cards initially. Both the pl 1 dealer may dray, additional cards (called “taking a hit”) in order to improve th f cither the playe or dealer takes a hit and the value of the hand exceeds 2 P r dealer is said to have gone broke and loses. Face cards and tens count 10 poinf t nted as | or 11, and all other cards count at their face value. The d hat the player must decide on whether to take a hit first. The player wh r 21 goes broke and loses, even if the dealer later goes brok or 1 has 16 and draws any card with a value higher than a 5, the pla For this re son, players will often decide not to take a hit when th or greater The dealer's hand is dealt with one card up an ayer’s deci. sion of whether to take a hit is based on knowledg gambling professional asks you to help determine the prob: e dealer's hand given different up cards. House rules at casir nue to take a hit until the dealer’s hand reaches a value of d Markoy processes, you suggest that the dealer's process of tak a Markov process with absorbing states. Managerial Report Prepare a report for the professional gambler that summariz Include the following 1. At some casinos, the dealer is required to stay (stop taking hits) when the dealer hand reaches soft or hard 17. A hand of soft 17 is one including an ace that may be counted as 1 or 11. In all casinos, the dealer is required to stay with soft 18,19, 20, or 21. For each possible up card, determine the probability that the ending value of the dealer's hand is 17, 18, 19, 20, 21, or broke, 2. At other casinos, the dealer is required to take a hit on soft 17, but must stay on all other hands with a value of 17, 18, 19, 20, or 21. For this situation, determine the probability of the ending value of the dealer's hand. 3. Comment on whether the house rule of staying on soft 17 or hitting on soft 17 appears better for the player

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Uploaded ImageCase Problem Dealer’s Absorbing State Probabilities in Blackjack The game of blackjack (sometimes called “21) is a popular casino game. The goal js 1, have a hand with a value of 21 or as close to 21 as possible without exceeding 21, Tp, player and the dealer are each dealt two cards initially. Both the pl 1 dealer may dray, additional cards (called “taking a hit”) in order to improve th f cither the playe or dealer takes a hit and the value of the hand exceeds 2 P r dealer is said to have gone broke and loses. Face cards and tens count 10 poinf t nted as | or 11, and all other cards count at their face value. The d hat the player must decide on whether to take a hit first. The player wh r 21 goes broke and loses, even if the dealer later goes brok or 1 has 16 and draws any card with a value higher than a 5, the pla For this re son, players will often decide not to take a hit when th or greater The dealer's hand is dealt with one card up an ayer’s deci. sion of whether to take a hit is based on knowledg gambling professional asks you to help determine the prob: e dealer's hand given different up cards. House rules at casir nue to take a hit until the dealer’s hand reaches a value of d Markoy processes, you suggest that the dealer's process of tak a Markov process with absorbing states. Managerial Report Prepare a report for the professional gambler that summariz Include the following 1. At some casinos, the dealer is required to stay (stop taking hits) when the dealer hand reaches soft or hard 17. A hand of soft 17 is one including an ace that may be counted as 1 or 11. In all casinos, the dealer is required to stay with soft 18,19, 20, or 21. For each possible up card, determine the probability that the ending value of the dealer's hand is 17, 18, 19, 20, 21, or broke, 2. At other casinos, the dealer is required to take a hit on soft 17, but must stay on all other hands with a value of 17, 18, 19, 20, or 21. For this situation, determine the probability of the ending value of the dealer's hand. 3. Comment on whether the house rule of staying on soft 17 or hitting on soft 17 appears better for the player

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Asked by: Umesh
Created at: 2025-08-01 22:30:56

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Certainly! Here is a summary and guidance for solving the **Dealer's Absorbing State Probabilities in Blackjack** case problem based on the image you provided: --- ### **Summary of the Problem** You're asked to model the dealer's process of taking hits in Blackjack as a **Markov process with absorbing states**. The goal is to: - Calculate the probability that the dealer ends up with each possible final hand value (17, 18, 19, 20, 21, or "broke"/bust), **given different up cards** and under different house rules regarding "soft 17". - Compare two common casino rules: 1. **Dealer stays on soft 17** 2. **Dealer hits on soft 17** --- ### **Managerial Report: Steps and Guidance** #### **1. Probability Calculation for Each Ending Value (Stay on Soft 17 Rule)** - **State Definition**: Each state represents a possible total value of the dealer's hand, with absorbing states being 17, 18, 19, 20, 21, and bust. - **Transitions**: From each non-absorbing state, the probability of moving to another state is determined by the chance of drawing each possible card (usually assuming an infinite deck or standard deck distribution). - **Soft 17 Rule**: The dealer **stays** on both soft and hard 17, as well as higher values (18, 19, 20, 21). - **Method**: Set up the Markov chain transition matrix, solve for the probability of ending in each absorbing state starting from each possible up card. #### **2. Probability Calculation for Each Ending Value (Hit on Soft 17 Rule)** - **Modified Rule**: Now, the dealer **hits** on soft 17, but stays on hard 17, 18, 19, 20, 21. - **Adjustment**: Update your Markov chain so that soft 17 is no longer an absorbing state; the dealer draws another card from this state. - **Method**: Adjust the transition matrix and solve for the new set of probabilities. #### **3. Comment on House Rule Favorability** - Compare the two sets of probabilities from above. - In general: - **Staying on soft 17** tends to **favor the player** slightly, because the dealer has a higher chance of ending on a weaker hand. - **Hitting on soft 17** increases the dealer’s chances of improving their hand, and therefore **favors the house**. --- ### **Suggested Structure for Your Managerial Report** #### **Introduction** - Briefly describe the rules and the two scenarios. #### **Analysis and Results** - **Part 1:** Present the probability table for each ending state (for each up card) under the "stay on soft 17" rule. - **Part 2:** Present the probability table for each ending state under the "hit on soft 17" rule. - **Part 3:** Discuss the impact on the player/house and which rule is more favorable to the player. --- ### **Tip for Calculations** - **Absorbing Markov Chain**: The key is to set up a transition probability matrix. The absorbing states will be 17, 18, 19, 20, 21, and bust. For each non-absorbing state, calculate the probability of moving to each other state by drawing a card. - **Software**: You can use Excel, Python, or even by hand for small examples. --- #### **Sample Table Format for Results (for each up card):** | Ending Hand | Probability (Stay on S17) | Probability (Hit on S17) | |-------------|---------------------------|--------------------------| | 17 | ... | ... | | 18 | ... | ... | | 19 | ... | ... | | 20 | ... | ... | | 21 | ... | ... | | Bust | ... | ... | --- If you need help with the actual calculations or setting up the Markov chain, let me know!

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