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Math Help - game theory,transition matrix

  1. #1
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    game theory,transition matrix

    1.a gambler with 2 dollar makes a series of one-dollar bets. His probability of winning one dollar is 2/5. He either wins one dollar or loses one dollar, he decides to quit playing as soon as he gains 4 dollars or loses 2 dollars.
    Let Xn be the amoun the Gambler is having at time n.
    Write down the one-step transition matrix for this process.

    2. Suppose 2 black and 2 white balls are placed in 2 boxes so that each box contains 2 balls. the number of the black balls in the first box is the state of the system. From each of the boxes, one ball is selected at random and the 2 balls are put back in the opposite boxes. Treating this process as a markov chain, find the transition matrix P.
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  2. #2
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    Quote Originally Posted by abiola View Post
    1.a gambler with 2 dollar makes a series of one-dollar bets. His probability of winning one dollar is 2/5. He either wins one dollar or loses one dollar, he decides to quit playing as soon as he gains 4 dollars or loses 2 dollars.
    Let Xn be the amoun the Gambler is having at time n.
    Write down the one-step transition matrix for this process.
    The possible state of the game is that the player has 0, 1, 2, ..., 6 dollars.
    The 0 and 6 dollar states are absorbing.

    So lets represent the game after n plays as a column vector the i th element being the probability that the game is in sate i. Then after the n th play the probabilities satisfy::


     <br />
p(i,n)=<br />
\begin{cases}<br />
p(0,n-1)+(3/5)p(1,n-1), & i=0\\<br />
p(6,n-1)+(2/5)p(5,n-1), & i=6\\<br />
(2/5)p(i-1,n-1)+(3/5)p(i+1,n-1),& \text{otherwise}<br />
\end{cases}<br />

    which give the 1-step transitions form which you should be able to calculate the 1-step transition matrix.

    CB
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