Jim is a gambler who pays his bills by spending weekends in Las Vegas.
Jim is a gambler who pays his bills by spending weekends in Las Vegas. Lately, he’s been playing 21 at a table that returns cards to the deck and reshuffles them all before each hand. As he has a fixed policy in how he plays, his probability of winning a particular hand remains constant, and is independent of all other hands. There is a wrinkle, however; the dealer switches between two decks (deck # 2 is more unfair to Jim than deck # 1), depending on whether or not Jim wins. Jim’s wins and losses can be modeled via the transitions of the following Markov chain, whose states correspond to the particular deck being used (see Figure 1).
(a) Model the problem as a DTMC. Clearly define the states and write out the transition matrix.
(b) What is Jim’s long-term probability of winning?
Given that Jim loses and the dealer is not occupied with switching the decks, with probability 2 8 the dealer looks away for one second and with probability 1 8 the dealer looks away for two seconds, independently of everything else. When this happens, Jim secretly inserts additional cards into both of the dealer’s decks, transforming the decks into types 1A & 2A (when he has 1 second) or 1B & 2B (when has has 2 seconds). Jim slips cards into the decks at most once. The process can be described by the transition graph in Figure 2. Assume in all future problems that play begins with the dealer using deck # 1.
(c) Model the problem as a DTMC. Clearly define the states and write out the transition matrix. (d) What is the probability of Jim eventually playing with decks 1A and 2A? (e) What is Jim’s long-term probability of winning? (f) What is the expected time (as in number of hands) until Jim slips additional cards into the deck?
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