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Math Help - Markov Chain coin toss

  1. #1
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    Markov Chain coin toss

    Coin 1 has probability of coming up heads with probability 0.7.
    Coin 2 has probability of coming up heads with probability 0.6.

    If we flip a coin today and it comes up heads, we will be flipping coin 1 tomorrow.

    If we flip a coin today and it comes up tails ,we will be flipping coin 2 tomorrow.

    if the coin initially flipped is equally likely to be coin 1 or coin 2, what is probability that the coin flipped on the third day after the initial flip is coin 1?


    I'm having trouble setting my matrix, I was pretty sure that my matrix would look like:

    \begin{bmatrix}0.7 & 0.3\\0.6& 0.4\end{bmatrix}

    but apparently this is wrong, since I have to consider the fact that both coins are equally likely to be flipped originally. Therefore you would have:

    0.5 \times 0.7 =0.35 of obtaining a head using coin one initially.
    0.5 \times 0.3 =0.15 of obtaining tails using coin one initially.
    0.5 \times 0.6=0.3 of obtaining a head using coin two initially.
    0.5 \times 0.4=0.2 of obtaining tails using coin two initially.

    The problem is that if I make a matrix using these values, they won't sum up to one; so I'm a little lost on what to do.
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  2. #2
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    Quote Originally Posted by lllll View Post
    Coin 1 has probability of coming up heads with probability 0.7.
    Coin 2 has probability of coming up heads with probability 0.6.

    If we flip a coin today and it comes up heads, we will be flipping coin 1 tomorrow.

    If we flip a coin today and it comes up tails ,we will be flipping coin 2 tomorrow.

    if the coin initially flipped is equally likely to be coin 1 or coin 2, what is probability that the coin flipped on the third day after the initial flip is coin 1?


    I'm having trouble setting my matrix, I was pretty sure that my matrix would look like:

    \begin{bmatrix}0.7 & 0.3\\0.6& 0.4\end{bmatrix}

    but apparently this is wrong, since I have to consider the fact that both coins are equally likely to be flipped originally. Mr F says: The transition matrix is {\color{red}\begin{bmatrix}0.6 & 0.7\\0.4& 0.3\end{bmatrix}}. The initial state matrix is {\color{red}\begin{bmatrix}0.5 \\ 0.5 \end{bmatrix}}. The final state matrix is {\color{red}\begin{bmatrix}Pr(C1) \\ Pr(C1') \end{bmatrix}}.

    [snip]
    ..
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  3. #3
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    Quote Originally Posted by lllll View Post
    I'm having trouble setting my matrix, I was pretty sure that my matrix would look like:

    \begin{bmatrix}0.7 & 0.3\\0.6& 0.4\end{bmatrix}
    It seems there are different conventions to define transition matrices. I am used to defining the transition matrix as P=(p(i,j))_{i,j} (the probability p(i,j) to go from state i to state j lies on row i and column j). In this case, your transition matrix is correct. If the initial probability distribution is \begin{bmatrix}0.5 & 0.5\end{bmatrix}, then after one step it becomes: \begin{bmatrix}0.5 & 0.5\end{bmatrix}\times \begin{bmatrix}0.7 & 0.3\\0.6& 0.4\end{bmatrix}. And, on the third day, the distribution is \begin{bmatrix}0.5 & 0.5\end{bmatrix}\times \begin{bmatrix}0.7 & 0.3\\0.6& 0.4\end{bmatrix}^2 (the first component of the vector you get after computation is the probability that coin 1 is flipped on the third day).
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    Quote Originally Posted by Laurent View Post
    It seems there are different conventions to define transition matrices.
    I am so glad that this has been noted by someone else. It has bothered me in at least one other post. I do not know of a textbook commonly used in North America that gives the transition matrix as ‘column’ driven. Can someone point me to a text that uses the column convention?
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    Quote Originally Posted by Plato View Post
    I am so glad that this has been noted by someone else. It has bothered me in at least one other post. I do not know of a textbook commonly used in North America that gives the transition matrix as ‘column’ driven. Can someone point me to a text that uses the column convention?
    I can't off-hand but I can say that in some parts of Australia this convention is taught at secondary and tertiary levels.
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    Quote Originally Posted by Plato View Post
    I am so glad that this has been noted by someone else. It has bothered me in at least one other post. I do not know of a textbook commonly used in North America that gives the transition matrix as ‘column’ driven. Can someone point me to a text that uses the column convention?
    You are no doubt referring to my previous random walk post. My professor instructed us in class to use a column driven transitional matrix. Oddly enough, the textbook for the class uses rows.
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