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Math Help - Probability of winning prize pool

  1. #1
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    Probability of winning prize pool

    You pick 6 numbers from 52. If all 6 of your numbers are picked you share a grand prize pool. If they draw 5 of your #'s you share a second prize pool. If the draw 4 of your #'s you share a third prize pool.

    What is the probability you will share in the 1st prize pool?
    Since order doesnt matter I found there are 2035820 different combinations, i think thats right.
    So is the probability just 6 over that number?
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  2. #2
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    Hello, mightydog78!

    You pick 6 numbers from 52.
    If all 6 of your numbers are picked, you share a Grand Prize pool.
    If they draw 5 of your numbers, you share a Second Prize pool.
    If they draw 4 of your numbers, you share a Third Prize pool.

    What is the probability you will share in the Grand Prize pool?

    Since order doesn't matter, I found there are 20,358,520 different combinations.

    So is the probability just 6 over that number? . No

    There is only one combination that contains all six winning numbers.

    . . P(\text{Grand Prize}) \;=\;\dfrac{1}{20,\!358,\!520}


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    There are 6 Winning Numbers and 46 Others.



    To win the Second Prize, you must have 5 of the 6 Winners
    . . and 1 of the 46 Others.

    There are: . \displaystyle {6\choose5}{46\choose1} \:=\:6\cdot46 \:=\:276\text{ ways.}

    . . P(\text{2nd Prize}) \;=\;\dfrac{276}{20,\!358,\!520} \;=\;\dfrac{69}{5,\!089,\!630}



    To win the Third Prize, you must have 4 of the 6 Winners
    . . and 2 of the 46 Others.

    There are: . \displaystyle {6\choose4}{46\choose2} \:=\:15\cdot1035 \:=\:15,\!525\text{ ways.}

    . . P(\text{3rd Price}) \;=\;\dfrac{15,\!525}{20,\!358,\!520} \;=\;\dfrac{3,\!105}{4,\!071,\!704}

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