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Math Help - Dice Probability

  1. #1
    Jen
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    Dice Probability

    A die is rolled 1000 times. Show that the probability that the sum of the numbers shown is 1100 is the same as the probability that the sum of the numbers shown is 5900.

    Not really sure how to go about doing this.
    Some hints would be nice.

    I would appreciate it if I wasn't given the answer all written out. I just want an idea of where to start.

    thank you in advance
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  2. #2
    Moo
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    Hello,

    Hmm I'll give it some thought...

    I think the main idea for this is :
    \mathbb{P}(X_n=i)=\mathbb{P}(X_n=7-i) \quad \forall i\in\{1,2,3,4,5,6\}, and where X_n is the score of the n-th dice.
    But I haven't tried to use it for a formal proof yet.
    Last edited by Moo; May 13th 2009 at 09:29 AM. Reason: 7, not 6...
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  3. #3
    Moo
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    Okay, let me rephrase that. (note : I edited my first version, where I put 6-X_n instead of {\color{red}7}-X_n) :

    X_n and 7-X_n follow the same distribution.
    That is \forall i\in\{1,2,3,4,5,6\} ~,~ \mathbb{P}(X_n=i)=\mathbb{P}(7-X_n=i)

    And here is my try for a formal proof... I'll ask my teacher for a more beautiful one, if you're interested.

    Spoiler:

    Let's study the n-tuple X=(X_1,\dots,X_n)
    This follows the same distribution as X'=(7-X_1,\dots,7-X_n)

    This means that for any measurable application h, we have \mathbb{E}(h(X))=\mathbb{E}(h(X')) \quad {\color{red}\star}

    Let :
    h\equiv h_a~:~ \mathbb{R}^n\rightarrow \mathbb{R}
    h_a((x_1,\dots,x_n))=\begin{cases} 1 \quad \text{if }x_1+\dots+x_n=a \\ 0 \quad \text{otherwise} \end{cases}

    It is obvious that :
    \mathbb{E}(h_a(X))=\mathbb{P}(X_1+\dots+X_n=a)
    and
    \mathbb{E}(h_a(X'))=\mathbb{P}(7-X_1+\dots+7-X_n=a)=\mathbb{P}(X_1+\dots+X_n=7n-a)

    By \color{red} \star, we get \boxed{\mathbb{P}(X_1+\dots+X_n=a)=\mathbb{P}(X_1+  \dots+X_n=7n-a)}

    Which is a generalization of your situation (n=1000, a=1100)
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  4. #4
    Jen
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    That was a beautiful proof Moo! Thank you.


    I like the fact that you labeled it a "spoiler".
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  5. #5
    MHF Contributor matheagle's Avatar
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    It's a symmetric distribution with a mean of 3500.
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