Results 1 to 5 of 5

Math Help - Transition Matrix - homework check

  1. #1
    Junior Member
    Joined
    Nov 2008
    Posts
    53

    Transition Matrix - homework check

    52. Three people -- John, Joan, and Kim -- throw a ball to each other. There is a probability of 1/3 that John will throw the ball to Joan. There is a probability of 1/2 that Joan will throw the ball to Kim. There is a probability of 1/4 that Kim will throw the ball to John.

    b) Assuming that the ball starts with Joan, what is the probability that she will have it back after 2 throws?
    _____ ___ ___
    [__0 _____ 1/3 ____ 2/3 ]2 (this matrix squared)...
    [ _ 1/2 ____ 0 ______ 1/2 ]
    [__ 1/4 ____ 3/4 _____ 0 ]

    ..which gives you
    [__1/3_____ 1/6 ____ 1/6 ]
    [ _ 1/8 ____ 13/24___ 1/3 ]
    [__ 3/8 ____ 1/12_____13/24 ]

    Therefore, there is a 13/24 probability that Joan will have the ball.

    Second part of question.. Assuming that the ball starts with Kim, what is the probability that Joan will have it after 3 throws??

    THANKS
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,547
    Thanks
    539
    Hello, cnmath16!

    Three people: John, Joan, and Kim, throw a ball to each other.
    There is a probability of 1/3 that John will throw the ball to Joan.
    There is a probability of 1/2 that Joan will throw the ball to Kim.
    There is a probability of 1/4 that Kim will throw the ball to John.

    a) Assuming that the ball starts with Joan,
    what is the probability that she will have it back after 2 throws?

    \begin{array}{cccc} & \text{John} & \text{Joan} & \text{Kim} \\<br />
\text{John} & 0 & \frac{1}{3} & \frac{2}{3} \\ \\[-4mm]<br />
\text{Joan} & \frac{1}{2} & 0 & \frac{1}{2} \\ \\[-4mm]<br />
\text{Kim} & \frac{1}{4} & \frac{3}{4} & 0 \end{array}\quad=\quad A


    Then: . A^2 \;=\;\begin{pmatrix}\frac{1}{3} & \frac{1}{2} & \frac{1}{6} \\ \\[-4mm]<br />
\frac{1}{8} & {\color{blue}\frac{13}{24}} & \frac{1}{3} \\ \\[-4mm]<br />
\frac{3}{8} & \frac{1}{12} & \frac{13}{24} \end{pmatrix}


    Therefore, there is a \tfrac{13}{24} probability that Joan will have the ball after 2 throws.

    Correct!



    b) Assuming that the ball starts with Kim,
    what is the probability that Joan will have it after 3 throws?
    We want A^3.

    A^3 \:=\:A^2\cdot A \:=\:\begin{pmatrix}\frac{1}{3} & \frac{1}{2} & \frac{1}{6} \\ \\[-4mm] \frac{1}{8} & \frac{13}{24} & \frac{1}{3} \\ \\[-4mm] \frac{3}{8} & \frac{1}{12} & \frac{13}{24}\end{pmatrix} \begin{pmatrix}0 & \frac{1}{3} & \frac{2}{3} \\ \\[-4mm] \frac{1}{2} & 0 & \frac{1}{2} \\ \\[-4mm] \frac{1}{4} & \frac{3}{4} & 0 \end{pmatrix} . =\:\begin{pmatrix}\frac{7}{24} & \frac{17}{72} & \frac{17}{36} \\ \\[-4mm] \frac{17}{48} & \frac{7}{24} & \frac{17}{48} \\ \\[-4mm] \frac{17}{96} & {\color{blue}\frac{17}{32}} & \frac{7}{24} \end{pmatrix}

    If the ball starts with Kim, Joan has the ball in 3 throws with probability \frac{17}{32}

    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2008
    Posts
    53
    Thanks very much for your reply..
    One quick question.


    For A^2, you have 1/2 in the top middle of the matrix. Is this a typo? Should the first row be 1/3 1/6 1/6... Not 1/3 1/2 1/6?

    And will this change the final answer?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,977
    Thanks
    1121
    1/2 is correct. It is (0)(1/3)+ (1/3)(0)+ (2/3)(3/4)= (1/3)(3/2)= 1/2.

    You can trust Soroban!
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,547
    Thanks
    539
    Hello, cnmath16!


    A product of two transition matrices is another transition matrix.

    This means: every row must add up to one (1).

    That's how I checked my work.

    (And it took me several tries to get it right!)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Transition Matrix
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: December 2nd 2009, 12:49 PM
  2. Transition Matrix
    Posted in the Statistics Forum
    Replies: 1
    Last Post: September 5th 2009, 05:37 PM
  3. transition matrix
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: April 23rd 2009, 11:36 PM
  4. transition matrix
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: March 30th 2009, 06:20 PM
  5. Transition Matrix, check answer please.
    Posted in the Pre-Calculus Forum
    Replies: 0
    Last Post: August 8th 2006, 08:35 AM

Search Tags


/mathhelpforum @mathhelpforum