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    I am doing a question on markov chains but I am already stuck on how to set the transition matrix up:

    A Quiz machine is in a pub, the probability of winning depends only on the two previous results.

    Result at n-2: L L W W
    Result at n-1: L W L W
    Prob of win at n:a 0.5 0.5 1-b


    create a transition matrix of a 4 state markov chain. To do this take the states at time n to be the four possible pairs of results at n-1 and n?
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    I'm afraid I can't remember how to write matrices in latex so will have to just give a quick reply.
    The four states that you are interested in are WW, WL, LW and LL

    If your previous state was WW then the probability of LW or LL after the next go is clearly 0, the prob of WW is 1-b (the prob that you win the next go given WW) and the prob is WL is b (the prob that you lose the next go given WW).

    Similar approach for the others will allow you to complete your matrix- you should have two 0 probabilities in each row as two of the outcomes will always be impossible.
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    Ah so like this for LL LW WL WW:

    \[ \left( \begin{array}{cccc}

1-a & 0 & 0.5 &0 \\

a & 0 & 0.5& 0\\

0&0.5&0&b\\

0 & 0.5 & 0 & 1-b\end{array} \right)\]

    ?
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    Exactly, but it should be the other way round

    \[ \left( \begin{array}{cccc}

1-a & a & 0 &0 \\

0 & 0 & 0.5& 0.5\\

0.5&0.5&0&0\\

0 & 0 & b & 1-b\end{array} \right)\]

    The row represents your current state and the columns represent the state after the next turn.
 
 
 
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