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Probability 2 Dice Question

  1. Feb 12, 2014 #1
    1. The problem statement, all variables and given/known data

    A pair of dice is rolled once, what is the probability that neither a doublet nor the sum of 10 will appear

    2. Relevant equations

    P(A) = 1 - P(A')

    Demorgans law
    (AUB)c = Ac ∩ B c


    3. The attempt at a solution

    I know how to do the solution through brute force and listing out all the possible scenarios:
    (1,1), (1,2), (1,3) ...... (6,6) and doing it like that. But i wanted a more algebraic approach so I tried this:

    Let A = is a doublet
    Let B = sum of dices is 10

    so then what I am looking for is

    P(Ac U Bc)

    and applying demorgans

    P(Ac U Bc) = P(A∩B )c

    using the law that Ac = 1 - A

    I got the final easy equation

    1 - P(A ∩ B )

    solving for this

    P(A) = 6/36 ...........(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)
    P(B) = 3/36 ............(4,6), (5,5),(6,4)

    plug in numbers

    1 - (6/36)*(3/36) != the answer of 7/9

    Apparently, math lies, j.k. Can someone let me know what i am doing wrong?
     
  2. jcsd
  3. Feb 12, 2014 #2

    Ray Vickson

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    Use inclusion-exclusion properly:
    [tex] \text{P}(A \cup B) = \text{P}(A) + \text{P}(B) - \text{P}(A \cap B) = 6/36 + 3/36 - 1/36.[/tex]
     
  4. Feb 12, 2014 #3
    Thats a clean method.

    Can I ask you why deMorgans doesn't work in this scenario?
     
  5. Feb 12, 2014 #4

    haruspex

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    Not so.
     
  6. Feb 12, 2014 #5

    Ray Vickson

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    It does work, and it is essentially what I used:
    [tex] \text{desired answer} = \text{P}(A^c \cap B^c) = 1 - \text{P}(A \cup B)[/tex]
    If you evaluate ##\text{P}(A \cup B)## correctly you will get the right answer.
     
  7. Feb 12, 2014 #6
    OHHHHH...my initial set up was wrong

    this: P(Ac∩Bc)

    and not this: P(AcUBc)

    the phrase "neither a nor b"

    confused me. I thought nor implied OR not AND.
     
  8. Feb 12, 2014 #7

    haruspex

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    "neither ... nor ..." is the same as "not ... and not ...".
     
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