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Math Help - probability

  1. #1
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    probability

    A and B plays chess everyday . The probability that A beats B in any game is twice the probability that B beats A . The probability that B beats A is three times the probability that the game ends in a draw . What is the probability that the game ends in a draw ?

    My working :
    P(A wins ) = 2(1- P(A wins ))
    P(A wins ) = 2/3

    P(B wins) = 3(1-P(A wins)-P(B wins))
    P(B wins) = 3(1-2/3-P(B wins ))
    P(B wins )=1/4

    P(draw)=1-2/3-1/4=1/12 --- my answer

    My answer is wrong . I wonder where my mistake is .....
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  2. #2
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    mathaddict,

    Quote Originally Posted by mathaddict View Post
    A and B plays chess everyday . The probability that A beats B in any game is twice the probability that B beats A . The probability that B beats A is three times the probability that the game ends in a draw . What is the probability that the game ends in a draw ?

    My working :
    P(A wins ) = 2(1- P(A wins ))
    P(A wins ) = 2/3
    P(B wins) is not equal to 1- P(A wins). This is because you have to also consider the possibility of a draw.

    Thus P(B wins) = 1- P(A wins ) - P(draw).
    So work out the exercise again, carefully...

    Good Luck
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  3. #3
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    Re :

    THanks Isomorphism ,

    My second attempt :

    P(A wins ) = 2[1-P(B wins)-P(draw)]
    P(A wins) = 2-2P(B wins)-2P(draw) ------ 1st equation

    P(B wins)= 3P(draw) ----- 2nd equation

    P(draw) = 1-P(A wins)-P(B wins ) ---- 3rd equation

    But after simplifying , i got P(draw)=1/4
    My answer is wrong according to the book . Did i get the equations right in the first place ? Thanks for any help ..
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  4. #4
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    Quote Originally Posted by mathaddict View Post
    THanks Isomorphism ,

    My second attempt :

    P(A wins ) = 2[1-P(B wins)-P(draw)]
    P(A wins) = 2-2P(B wins)-2P(draw) ------ 1st equation

    P(B wins)= 3P(draw) ----- 2nd equation

    P(draw) = 1-P(A wins)-P(B wins ) ---- 3rd equation

    But after simplifying , i got P(draw)=1/4
    My answer is wrong according to the book . Did i get the equations right in the first place ? Thanks for any help ..
    Let Pr(B beats A) = Pr(A loses) = p.

    Given: Pr(A beats B) = Pr(A wins) = 2p.

    Given: Pr(A and B draw) = p/3.

    p + 2p + (p/3) = 1.

    Solve for p.

    I get Pr(Draw) = 1/10.
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  5. #5
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    Re :

    Thanks Mr F , but the answer given is 1/9
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  6. #6
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    Let p_d be the probability of a draw, p_b probability of B winning and p_a the probability of A winning.

    Then:

    3 p_d=p_b

    2 p_b=p_a=6 p_d

    and of course:

     <br />
p_a+p_b+p_d=1<br />

    or:

    6 p_d+3p_d+p_p=1

    so:

    p_d=1/10

    as Mr Fantastic has.

    Now the ony way out I can see is if the question asked for the odds of a draw since the odds are the ratio of favourable to unfavourable outcomes the odds are 1:9.

    .
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  7. #7
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    Quote Originally Posted by Constatine11 View Post
    Let p_d be the probability of a draw, p_b probability of B winning and p_a the probability of A winning.

    Then:

    3 p_d=p_b

    2 p_b=p_a=6 p_d

    and of course:

     <br />
p_a+p_b+p_d=1<br />

    or:

    6 p_d+3p_d+p_p=1

    so:

    p_d=1/10

    as Mr Fantastic has.

    Now the ony way out I can see is if the question asked for the odds of a draw since the odds are the ratio of favourable to unfavourable outcomes the odds are 1:9.

    .
    Which shows why it's important to understand the difference between the probability of an event as represented by:

    1. A number between zero and 1 inclusive .

    2. Odds of the form a : b

    3. Percentage

    etc.
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