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Discrete-Time
Markov Chain |
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Absorbing Probabilities |
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For a matrix with absorbing states, the
Absorbing worksheet computes for each transient
state the probability that the system will terminate in a particular
absorbing state. To illustrate absorbing states, we provide
a Markov Chain that describes the game of Craps. As shown below
the transition matrix describes a system with two absorbing
states,
Win and Lose. The state First is the
state when the game begins. The states P4 through P10 describe
the situation of throwing the numbers 4 through 10 on the first
roll. All states except Win and Lose are
transient. The gambler is interested in computing the
probabilities of Winning or Losing and how they depend on
his current state? The transition matrix is shown below. |
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The absorbing
state analysis determines the information in the table below.
The first row of the table shows the probability of winning
and losing starting from state First, that is, the beginning
of the game. The chance of losing is slightly greater than
the chance of winning. The other rows show the probabilities
of winning and losing if the gambler rolls one of the point
numbers. The best bet for the gambler is a point of 6 or 8,
while the worst is a point of 4 or 10. |
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