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Math Help - Probability question with dice

  1. #1
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    Probability question with dice

    Situation: When you throw 4 dice at once, the chances all the dice will be divisible by 3 is 1/16

    Question: Prove the amount of dice you throw is 4.

    Can somebody please explain how the amount of dice is 4?????

    Here's how I tried understanding: When you throw 4 dice, there are 5 possible situations you can get all the dice divisble by 3:

    1. 3,3,3,3
    2. 3,3,3,6
    3. 3,3,6,6
    4. 3,6,6,6
    5. 6,6,6,6

    Then I counted all the possible situations you can get when you have 4 dice
    (1,1,1,1; 1,1,1,2.....) and that adds up to 24.

    So I got that when you spin 4 dice the odds are 5/24 that all the dice will be divisble by 3.

    Can someone please help? Thanks
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  2. #2
    MHF Contributor Unknown008's Avatar
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    Re: Probability question with dice

    For 2, you also have 3, 3, 6, 3; 3, 6, 3, 3; 6, 3, 3, 3

    You have to consider each die separately.

    Same for the others

    The total number of possibilities for 1 die is 6. For 2 dice, it's 6*6, and for 4 dice, you have 6*6*6*6 = 1296 possibilities

    Ways to get all divisible by 3 is only when you get 3 and 6.
    There are 5 'groups' of them that you got.

    3, 3, 3, 3
    3, 3, 3, 6; 3, 3, 6, 3; 3, 6, 3, 3; 6, 3, 3, 3
    3, 3, 6, 6; 3, 6, 3, 6; 6, 3, 3, 6; 3, 6, 6, 3; 6, 3, 6, 3; 6, 6, 3, 3
    3, 6, 6, 6; 6, 3, 6, 6; 6, 3, 6, 6; 6, 6, 6, 3
    6, 6, 6, 6

    This makes 16 in all.

    The probability is then 16/1296

    Hm... are you sure the dice is a regular dice? I think if it were a tetrahedron with 3, 6 and two other numbers less than 6, it'd be getting 1/16.
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    Re: Probability question with dice

    i am sure it is a normal die. Maybe the book made a mistake.
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    Re: Probability question with dice

    Perhaps the dice are identical and we cannot distinguish between 3363 and 3633. But, I still do not get 1/16 with this assumption!
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    Re: Probability question with dice

    Maybe they made a mistake with the number 4. Maybe if you throw 3 or 5 dice you will get 1/16??
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    Re: Probability question with dice

    Quote Originally Posted by jayshizwiz View Post
    Maybe they made a mistake with the number 4. Maybe if you throw 3 or 5 dice you will get 1/16??
    The number of dice is neither 3 nor 5.
    If the dice were not identical, with 3 dice, the answer would be 1/27 and with 5 dice, the answer would be 1/243.
    If the dice were identical, with 3 dice, the answer would be 1/14 and with 5 dice, the answer would be 1/42.

    Here is a general formula:
    ** If 'n' non-identical dice are thrown, then the answer to your question is \inline\displaystyle \frac1{3^n}

    ** If 'n' identical dice are thrown, then the answer to your question is \inline \displaystyle \frac{n+1}{\displaystyle \sum_{k=1}^{n}\binom{6}{k}\binom{n-1}{k-1}} with the convention that \inline \displaystyle \binom{n}{k} = 0 whenever \inline \displaystyle k > n
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