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Probability: Dice Game

  1. Sep 22, 2009 #1
    1. The problem statement, all variables and given/known data
    Sarah and Thomas each roll a die. Whoever gets the higher number wins; if they both roll they same number, neither wins.

    a. What is the probability that Thomas wins?

    b. If Sarah, rolls a 3 what is the probability that she wins?

    c. If Sarah rolls a 3 what is the probability that Thomas wins?

    d. If Sarah wins what is the probability that Thomas rolled a 3?

    e. If Sarah wins what is the probability that Sarah rolled a three?


    2. Relevant equations

    P(A|B) = P(A[tex]\cap[/tex]B)/P(B)

    Bayes' Rule


    3. The attempt at a solution

    For a, is it as simple as they both at the beginning of the game have an equally likely chance of winning so 1/2?

    b) If she rolls a 3 then in order for her to win Thomas must roll a 1 or 2 so the probability that she wins is 1/3

    c) If she rolls a three then in order for Thomas to win he must roll a 4,5 or 6 so the probability that he wins is 1/2.

    d) If Sarah wins she must have rolled a 2,3,4,5, or 6. For Thomas to lose he must have rolled a 1,2,3,4 or 5. So the probability that he rolls a 3 is 1/5.

    e) If Sarah wins she must have rolled a 2,3,4,5, or 6. So the probability that she rolls a three is also 1/5.

    Is this correct reasoning? Also how do I actually use the relevant equations in this problem?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 23, 2009 #2

    lanedance

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    Homework Helper

    i don't think so, as there is also the potential no one wins...
    these sound right to me
    not sure i follow you reasoning on the last two

    the question revolves around the random variable, X which can take values {1,2,..,6}
    with P(X=n) = 1/6 for n in {1,2,..,6}

    for example a) is given the two independent random variables X1 & X2 with distributions as given above, what is P(X1 > X2)...

    however, as you were implying, due to symmetry, you know P(X2 > X1) = P(X1 > X2)
     
    Last edited: Sep 23, 2009
  4. Sep 23, 2009 #3

    CompuChip

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    Homework Helper

    For a), I think not. If you write out all the possibilities for example, you will see that in less than half of them Thomas actually wins.

    Similarly you over-simplified d) and e). Sarah winning does not mean that she rolled 2, 3, 4, 5 or 6 while Thomas got 1, 2, 3, 4 or 5. For example: if Sarah throws 3 and Thomas 4, then Thomas wins.
     
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