Results 1 to 10 of 10

Math Help - Marcov Chains

  1. #1
    Junior Member
    Joined
    Mar 2014
    From
    Columbus, Ohio
    Posts
    25

    Question Marcov Chains

    Lets say that you have 2 cups and in 1 of them there is an equal number of white and black say 100 white and 100 black. Lets say that in the other cup you add another 100 black to the 100 white and 100 black you put in there.

    Now lets say the black is 0 and white is 1.

    These are the transformations:
    white from 50/50 back to the same cup
    black from 50/50 go to 2/3 black 1/3 white
    black from 2/3 black back to same cup
    white from 2/3 black go to 50/50

    Now what is the probability of each of these outcomes?

    With 3 colors base 3
    with 4 colors base 4 and so on we can express bases to any number as long as we have that many colors.

    We would have for 3 these outcomes:
    white from 33/33/33 back to same cup
    black from 33/33/33 to 25/50/25
    red from 33/33/33 to 25/25/50
    black from 25/50/25 back to same cup
    red from 25/50/25 to 25/25/50
    white from 25/50/25 to 33/33/33
    red from 25/25/50 back to same cup
    white from 25/25/50 to 33/33/33
    black from 25/25/50 to 25/50/25

    Now for any number of colors we are going to have x^2 outcomes

    How would we figure out the probability of each outcome in the binary black/white, ternary black/white/red, quarternary black/white/red/orange, quinary black/white/red/orange/yellow, base 6 black/white/red/orange/yellow/green, base 7 black/white/red/orange/yellow/green/blue, octal black/white/red/orange/yellow/green/blue/indigo, nonary black/white/red/orange/yellow/green/blue/indigo/violet, and base 10 black/white/red/orange/yellow/green/blue/indigo/violet/gray when we have the colors assigned to these digits:
    black = 0
    white = 1
    red = 2
    orange = 3
    yellow = 4
    green = 5
    blue = 6
    indigo = 7
    violet = 8
    and
    gray = 9
    and for any base we get x^2 possible outcomes?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,612
    Thanks
    591

    Re: Marcov Chains

    Hey caters.

    I'm sorry but I don't understand your question: could you be more specific?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Mar 2014
    From
    Columbus, Ohio
    Posts
    25

    Re: Marcov Chains

    For any number of colors how are we going to figure out the probability of x^2 outcomes?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,607
    Thanks
    1574
    Awards
    1

    Re: Marcov Chains

    Quote Originally Posted by caters View Post
    For any number of colors how are we going to figure out the probability of x^2 outcomes?
    Will you give a clear example of that question?
    Give an example using four colors. Show us what an outcome looks like.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Mar 2014
    From
    Columbus, Ohio
    Posts
    25

    Re: Marcov Chains

    red from 20/20/20/40 go to 20/20/40/20

    Notice the first number is % black, the second number is % white, the third number is % red and the fourth number is % orange. There are 4 possible outcomes per cup because 1 cup has equal amounts of black,white,red,and orange but the others have 40 % of 1 color and 20% of the others, and the fact that you can draw any of the 4 colors. There are 4 cups because we put more beads of a certain color in the other 3. That gives us x^2 outcomes possible for any number of colors.

    I will give you another example.

    red from 25/25/25/25 go to 20/20/40/20
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,612
    Thanks
    591

    Re: Marcov Chains

    If you fix the number of states, you can derive a transition matrix and use that. Are you familiar with these matrices?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Mar 2014
    From
    Columbus, Ohio
    Posts
    25

    Re: Marcov Chains

    I am familiar with adding, subtracting, and multiplying matrices but I am not familiar with transition matrices
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,607
    Thanks
    1574
    Awards
    1

    Re: Marcov Chains

    Quote Originally Posted by caters View Post
    I am familiar with adding, subtracting, and multiplying matrices but I am not familiar with transition matrices
    That is what I suspect is a true statement,
    I also suspect that you have no idea what this question is all about.
    If that is correct, how can we help you?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member
    Joined
    Mar 2014
    From
    Columbus, Ohio
    Posts
    25

    Re: Marcov Chains

    I know that this question is about dependent probability. It is a true statement that I am familiar with operating with matrices but not with transition matrices.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,612
    Thanks
    591

    Re: Marcov Chains

    A transition matrix capture all of the probabilities. If you want the probabilities at the nth time step then all you have to do is calculate T^n where T is the transition matrix.

    You can do this symbolically as well if you don't have actual constant numbers. By calculating T^n you get all the probabilities from going to state 0 to state n.

    So basically once you get T, and if you want all probabilities at a particular time spot, then you calculate T^n (symbolically or numerically) and then just use that to get them.

    A simple 2x2 example (2 states) is the following:

    [0.75 0.25]
    [0.15 0.85]

    where P(A1=1|A0=1) = 0.75, P(A1=2|A1=1) = 0.15 and P(A1=2|A0=1) = 0.25.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Markov Chains
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: April 26th 2009, 08:17 PM
  2. Markov Chains
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: March 6th 2009, 09:34 AM
  3. Union of chains
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 21st 2009, 08:10 PM
  4. Markov Chains
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: November 17th 2008, 09:33 AM
  5. Decay Chains
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 12th 2007, 05:30 AM

Search Tags


/mathhelpforum @mathhelpforum