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 2 cards initially. Both the player and dealer may dray 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 first. 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 decision 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 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.

problems

1. AT some casino's, 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.