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Math Help - a dice game

  1. #1
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    a dice game

    A and B are playing a dice game with one dice. A starts the game first. 6 will win. What is
    the probability that A will win the game?

    I found the answer by the help of 1/1-n,using series and limit. The answer is 6/11. But there is another solution and I can't understand the logic of it.The solution is as follows:

    A starts first.The probability that A will win is "x".
    The probability that B will win is 5
    -----.x
    6
    x + (5/6)x= 1 x=6/11
    There is also another solution:
    x= (1/6)(1) + (5/6)(5/6)u

    x[1 - 25/36] = 1/6

    x(11/36) = 1/6

    x = 6/11
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  2. #2
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    Re: a dice game

    Quote Originally Posted by serhanbener View Post
    A and B are playing a dice game with one dice. A starts the game first. 6 will win. What is
    the probability that A will win the game?
    Here is another way.
    \displaystyle\frac{1}{6}\sum\limits_{k = 0}^\infty  {{{\left( {\frac{{25}}{{36}}} \right)}^k}}  = ?
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  3. #3
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    Re: a dice game

    Quote Originally Posted by Plato View Post
    Here is another way.
    \displaystyle\frac{1}{6}\sum\limits_{k = 0}^\infty  {{{\left( {\frac{{25}}{{36}}} \right)}^k}}  = ?
    Yes. I had also used this method.

    P(A) = 1/6 + (5/6)(5/6)(1/6) + (5/6)^4.(1/6) + (5/6)^6.(1/6) + ...

    = (1/6).[1 + (5/6)^2 + (5/6)^4 + (5/6)^6 + .... to infinity]

    1 1 36
    = (1/6)----------- = (1/6)--------- = (1/6).------
    1 - (5/6)^2 1 - 25/36 36-25

    P(A) = 6/11
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  4. #4
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    Re: a dice game

    I want to learn the logic of the solutions that I have mentioned in #".
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  5. #5
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    Re: a dice game

    In the second alternative solution I think they mean this. A either wins on first throw or wins but not on first throw. For first throw p=1/6. For not on first throw
    p=(5/6)(5/6)x (This is because after A and B have failed on first throw its like starting a new match so probability of A winning overall is still x)
    So x=(1/6)+(5/6)(5/6)x
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  6. #6
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    Re: a dice game

    I think you are right.Many Thanks.
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  7. #7
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    Re: a dice game

    Quote Originally Posted by serhanbener View Post
    A and B are playing a dice game with one dice.
    A starts the game first. 6 will win. What is the probability that A will win the game?
    Try this version (A starts, then B 2 in a row, then A 2 in a row....) :
    A B B A A B B .....
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  8. #8
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    Re: a dice game

    Quote Originally Posted by Wilmer View Post
    Try this version (A starts, then B 2 in a row, then A 2 in a row....) :
    A B B A A B B .....
    I am sorry. I couldn't understand.
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