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Conditional Dependent Probability

  1. Jan 2, 2010 #1
    My questions I am looking at is:

    We have twelve balls, four of which are white and eight are black. Three blindfolded players, A, B, and C draw a ball in turn, first A, then B, then C. The winner is the one who first draws a white ball. Assuming that each black ball is replaced after being drawn, find the ratio of the chances of each player.

    I do not have any background on dependent proability with replacement, only without.

    When making a probability tree with replacement would it look like this:

    ......PlayerA.................Player B...............Player C
    .........- ............................- ........................-
    .......- - - ....................... - - -....................- - -
    ......- - - -.......................- - - - .................- - - -
    .....b......w....................b.......w................b......w
    ...8/12...4/12.............8/13.....4/13...........8/14...4/14

    Please ignore the dots, it's the only way I could get my probability tree to look right.

    My logic is that player A has an advantage because he's going first. So to reduce player 2's chances I added one to the sample set to symbolize that a turn had already been taken. Am I right in my thinking?
     
    Last edited: Jan 2, 2010
  2. jcsd
  3. Jan 2, 2010 #2

    mathman

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    Science Advisor

    For the first round, A has a P of 1/3, B has a P of (2/3)1/3, while C has a P of (2/3)21/3. For each subsequent round, the ratio of their chances are the same. Thus their probabilities remain in the ratio 9:6:4.
    So P(A)=9/19, P(B)=6/19, P(C)=4/19.
     
  4. Jan 2, 2010 #3
    Thank you so much! That makes so much more sense then what I was trying to do!! Is there a way for me to rate your response as AWESOME!
     
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