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Math Help - Randome Walk Problem

  1. #1
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    Randome Walk Problem

    Peter has probability 1/4 of winning each game. Peter and Paul each bet $100 on each game. They each start with $400 and play until one of them goes broke. What is the probability that Paul goes broke?

    Answer from the book: 0.01220
    How do you do this problem>?

    I used this formula:
    Suppose Peter starts with s dollars and Paul starts with t-s dollars. Let P* be the probability that Peter wins all the money. Then
    P* = (1 -r^s) /(1-r^t)
    where r = q/p (p not equal q).

    Attempt:
    s = 400 (Given)
    t = 800 (from t-400= 400)
    p = 1/4 (Given)
    q = 3/4 (q=1-p) = (1-1/4)
    r = 3 (q/p)

    After that I substituted each value into the equation
    P* = (1 -r^s) /(1-r^t)
    P* = (1 -3^400) /(1-3^800)
    P* = 1.41742 E-191 (This is wrong)

    But when I tried s = 4 and t = 8
    P* = (1 -r^s) /(1-r^t)
    P* = (1 -3^4) /(1-3^8)
    P* = 1/82 = 0.012195. (which is correct).

    Can someone explain this to me? Why does my first attempt gives the wrong solution? When I tried another problem just like this (with s and t being single digits), it works well with direct substitution.
    Last edited by jjbrian; April 25th 2010 at 03:10 PM.
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  2. #2
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    Quote Originally Posted by jjbrian View Post
    Peter has probability 1/4 of winning each game. Peter and Paul each bet $100 on each game. They each start with $400 and play until one of them goes broke. What is the probability that Paul goes broke?

    Ans: 0.01220
    How do you do this problem>?

    I used this formula:
    Suppose Peter starts with s dollars and Paul starts with t-s dollars. Let P* be the probability that Peter wins all the money. Then
    P* = (1 -r^s) /(1-r^t)
    where r = q/p (p not equal q).

    Attempt:
    s = 400 (Given)
    t = 800 (from t-400= 400)
    p = 1/4 (Given)
    q = 3/4 (q=1-p) = (1-1/4)
    r = 3 (q/p)
    When you do this you assume that they start with $400 and play $1 each game. Obviously the longer the game is, the less probability that Peter has of winning.
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  3. #3
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    the information given (Peter and Paul each bet $100 on each game.They each start with $400 and play until one of them goes broke.) in the question is just for you to figure out the state space which is 1,2,3,4 in this case
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