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Math Help - Transition matrix problem

  1. #1
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    Transition matrix problem

    Market research shows that if a person buys Brand X, the probability that they will buy Brand X next time is 0.75. If a person buys Brand Y, the probability that they will buy B the next time is 0.55.
    Keith bought Brand X today.

    a) Write the initial probability matrix and the transition matrix
    b) what is the probability that keith will switch to Brand Y on his next purchase


    My first reaction was for initial matrix Po= [ 0.75, 0.55] ???

    for transition it would be
    0.75 0.25
    0.45 0.55


    Please help
    Last edited by terminator; September 22nd 2011 at 02:44 PM.
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  2. #2
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    Re: Transition matrix problem

    Quote Originally Posted by terminator View Post
    Market research shows that if a person buys Brand X, the probability that they will buy Brand X next time is 0.75. If a person buys Brand Y, the probability that they will buy B the next time is 0.55.
    Keith bought Brand X today.

    a) Write the initial probability matrix and the transition matrix
    b) what is the probability that keith will switch to Brand Y on his next purchase


    My first reaction was for initial matrix Po= [ 0.75, 0.55] ???
    The initial matrix is supposed to describe the initial situation. In this case, it should represent what Keith did today. What you are told about that is that Keith definitely bought Brand X, and definitely did not buy Brand Y. The numbers in the initial matrix should encode that information.
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    Re: Transition matrix problem

    The initial matrix is supposed to describe the initial situation. In this case, it should represent what Keith did today. What you are told about that is that Keith definitely bought Brand X, and definitely did not buy Brand Y. The numbers in the initial matrix should encode that information.

    So[ 0.50, 0.50] - initially?
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  4. #4
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    Re: Transition matrix problem

    No, the problem specifically says that "Keith bought Brand X today", not that he was equally likely to buy brand X or brand Y.
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    Re: Transition matrix problem

    [1,0] initial one then??

    0.75 0.25 - transition
    0.45 0.55

    If I multiply them I get 0.75 0.25

    The answer in my book is 0.4?
    Am I missing something?
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    Re: Transition matrix problem

    You don't really need the matrix to answer the second part (it would be useful if they asked about the probability of buying brand X or brand Y on the fifth, or seventh or 100th time). You are told that "if a person buys Brand X, the probability that they will buy Brand X next time is 0.75".

    Okay, Keith did buy brand x this time and so the probability he will buy it again the next time is 0.75. So either the text is wrong or you have quoted the exercise incorrectly.

    I note that A^7P_0 has an X component of .403 which rounds to .4. Is it possible that the question not asking for the next time he purchases this kind of product but the 6th time (since he has already bought once)?
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    Re: Transition matrix problem

    You don't really need the matrix to answer the second part (it would be useful if they asked about the probability of buying brand X or brand Y on the fifth, or seventh or 100th time). You are told that "if a person buys Brand X, the probability that they will buy Brand X next time is 0.75".

    Okay, Keith did buy brand x this time and so the probability he will buy it again the next time is 0.75. So either the text is wrong or you have quoted the exercise incorrectly.

    I note that A^7P_0 has an X component of .403 which rounds to .4. Is it possible that the question not asking for the next time he purchases this kind of product but the 6th time (since he has already bought once)?
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  8. #8
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    Re: Transition matrix problem

    You don't really need the matrix to answer the second part (it would be useful if they asked about the probability of buying brand X or brand Y on the fifth, or seventh or 100th time). You are told that "if a person buys Brand X, the probability that they will buy Brand X next time is 0.75".

    Okay, Keith did buy brand x this time and so the probability he will buy it again the next time is 0.75. So either the text is wrong or you have quoted the exercise incorrectly.

    I note that A^7P_0 has an X component of .403 which rounds to .4. Is it possible that the question not asking for the next time he purchases this kind of product but the 6th time (since he has already bought once)?
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