Problem DEALER’S ABSORBING STATE PROBABILITIES IN
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 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, play- ers will
often decide not to take a hit when the value of their hand is 12 or
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 pro- fessional 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 pro- cesses, you
suggest that the dealer’s process of taking hits can be modeled as
a Markov pro- cess with absorbing states.
a report for the professional gambler that summarizes your findings.
Include the following:
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.
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.
Comment on whether the house rule of staying on soft 17 or hitting on
soft 17 ap- p
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