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Math Help - Can someone please check this for me?

  1. #1
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    Can someone please check this for me?

    I am sure I know the answer to this question but it is very important that it is correct. Can someone please double check for me to make sure there are no mistakes?

    ----------------------------------------------------------------------------------------------------------------------
    QUESTION: If three (fair) dice are rolled what is the probability of rolling at most two matching numbers? what is the probability of rolling at least two matching numbers?

    ANSWER: total possible outcomes of rolling 3 dice = 6*6*6= 216

    two matching pair can be achieved by rolling
    [x, x, 1]
    [x, x, 2]
    [x, x, 3]
    [x, x, 4]
    [x, x, 5]
    [x, x, 6]
    where x = any number from 1-6

    Notice that at x=1 the first set is [1,1,1] which is more than two matching numbers. likewise at x=2 the second set would be [2, 2, 2] and so on for each number, x. therefore there are 5 possibilities in this case for each number in the set.

    six numbers each with five possible outcomes = 30 possible outcomes in this set

    Also two matching numbers could be achieved by

    [x, 1, x]
    [x, 2, x]
    [x, 3, x]
    [x, 4, x]
    [x, 5, x]
    [x, 6, x]

    or

    [1, x, x]
    [2, x, x]
    [3, x, x]
    [4, x, x]
    [5, x, x]
    [6, x, x]

    again each number, x has 5 possible outcomes giving us an additional 30 outcomes for each of these two sets.
    30*3 = 90
    divide by total possible outcomes
    90/216 = .4167

    AT LEAST two matching numbers includes the possiblity of rolling triples which can occur in six ways [1, 1, 1], [2, 2, 2], [3, 3, 3] and so on
    for a total probability of 6/216

    adding this to the previous answer we get 6/216 + 90/216 = 96/216 = .4444
    ----------------------------------------------------------------------------------------------------------------------

    again it is extremely important that these answers and the explanations are 100% correct. if anyone could double check the solution for me it would be greatly appreciated.

    Thank you
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, bm203!

    Your second answer is correct; your first answer is off.

    Let me show you a way to avoid making long lists . . .


    If three (fair) dice are rolled what is the probability of rolling:

    (a) at most two matching numbers?

    (b) at least two matching numbers?

    There are: . 6^3 = 216 possible outcomes.


    (a) At most two matching numbers. .This means 0 matching numbers,
    . . .1 matching number (whatever that means?!), or 2 matching numbers.

    We do not want 3 matching numbers.
    . . There are: 6 ways to have 3 matching numbers.

    Hence, there are: . 216 - 6 \,=\,210 ways to have at most 2 matching numbers.

    Therefore: . P(\text{at most 2 matches}) \:=\:\frac{210}{216} \:=\:\frac{35}{36}



    (b) At least 2 matching numbers.
    . . .This means: 2 matches or 3 matches.

    For 2 matches, there are 6 choices of for the value of the Pair.
    We want: \{X,\:X,\:\text{other}\} in some order.
    . . P(\text{2 matches}) \:=\:6\cdot {3\choose1}\left(\frac{1}{6}\right)^2\left(\frac{5  }{6}\right) \;=\;\frac{90}{216}

    For 3 matches: there are 6 choices for the value of the Triple.
    . . P(\text{3 matches}) \;=\;6\cdot\left(\frac{1}{6}\right)^6 \;=\;\frac{6}{216}


    Therefore: . P(\text{at least 2 matches}) \;=\;\frac{90}{216} + \frac{6}{216} \;=\;\frac{96}{216} \;=\;\frac{4}{9}

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  3. #3
    Newbie
    Joined
    Feb 2010
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    thank you. i did not mean to frame the question is such a way that would imply zero matches should be counted among the solutions. i will rephrase the first part of the question to say "at least two but not three matching numbers" or perhaps i'll simply make it a three part question.

    thanks again.
    Last edited by bm203; February 9th 2010 at 07:40 PM.
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