Discrete-Time Markov Chain - Absorbing Probabilities
 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. 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.

Operations Research Models and Methods
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by Paul A. Jensen
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