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Math Help - Lottery Probability

  1. #1
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    Smile Lottery Probability

    For a particular lottery, the winning numbers are selected by a machine that randomly chooses 5 table-tennis balls from among 45, numbered 1 to 45. The lottery pays off if you match 5, 4, or 3 of the numbers.
    a). How many different winning number combinations are there ?
    b). What is the probability that you will match all 5 of the winning numbers ?
    c). What is the probability that you will match exactly 4 of the 5 winning numbers ?

    Thank you.
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  2. #2
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    Lexington, MA (USA)
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    Hello, sahip!

    These are straight Combinations questions.
    Exactly where is your difficulty?


    For a particular lottery, the winning numbers are selected by a machine
    that randomly chooses 5 table-tennis balls from among 45, numbered 1 to 45.
    The lottery pays off if you match 5, 4, or 3 of the numbers.

    a) How many different winning number combinations are there?
    _{45}C_5 \:=\:{45\choose5} \:=\:\frac{45!}{5!\,40!} \;=\;1,\!221,\!759 possible winning combinations.


    b) What is the probability that you will match all 5 of the winning numbers?
    P(\text{5 numbers}) \;=\;\frac{1}{1,\!221,\!759}


    c) What is the probability that you will match exactly 4 of the 5 winning numbers?
    There are: . _5C_4 \:=\:{5\choose4} \:=\:5 ways to match 4 numbers.

    Your 5th number is from the other 40 non-winners: . 40 ways.

    . . Hence, there are: . 5\cdot40 \:=\:200 ways to match exactly 4 numbers.

    Therefore: . P(\text{exactly 4 numbers}) \;=\;\frac{200}{1,\!221,\!759}

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