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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.
    What is the probability that you will match exactly 4 of the 5 winning numbers ?

    Thank you.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by sahip View Post
    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.
    What is the probability that you will match exactly 4 of the 5 winning numbers ?

    Thank you.
    Suppose you match balls 1 to 4 but not 5, the probability of this is:

    \frac{1}{45}\times \frac{1}{44} \times \frac{1}{43}\times \frac{1}{42}\times \frac{40}{41}

    (this is the probability of matching ball 1, 2, 3, 4 and not matching ball 5)

    Now the final probability is 5 times this (the unmatched ball can be any of the five)

    CB
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    Suppose you match balls 1 to 4 but not 5, the probability of this is:

    \frac{1}{45}\times \frac{1}{44} \times \frac{1}{43}\times \frac{1}{42}\times \frac{40}{41}

    (this is the probability of matching ball 1, 2, 3, 4 and not matching ball 5)

    Now the final probability is 5 times this (the unmatched ball can be any of the five)

    CB
    Shouldn't you additionally multiply by 5! to account for the fact that the balls could be picked in 5! different orders?

    If so, then the answer would agree with an alternate approach I tried:

    Without loss of generality, we can suppose that the balls numbered 1 through 5 were randomly chosen, and we want the probability that exactly four match.

    {1,2,3,4,x}
    {1,2,3,x,5}
    {1,2,x,4,5}
    {1,x,3,4,5}
    {x,2,3,4,5}

    The above is to illustrate that there are 40*5 combinations for which exactly four match, so the probability is given by 40*5/C(45,5).
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  4. #4
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    Quote Originally Posted by undefined View Post
    Shouldn't you additionally multiply by 5! to account for the fact that the balls could be picked in 5! different orders?

    If so, then the answer would agree with an alternate approach I tried:

    Without loss of generality, we can suppose that the balls numbered 1 through 5 were randomly chosen, and we want the probability that exactly four match.

    {1,2,3,4,x}
    {1,2,3,x,5}
    {1,2,x,4,5}
    {1,x,3,4,5}
    {x,2,3,4,5}

    The above is to illustrate that there are 40*5 combinations for which exactly four match, so the probability is given by 40*5/C(45,5).
    Yes.

    CB
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  5. #5
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    Quote Originally Posted by sahip View Post
    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.
    What is the probability that you will match exactly 4 of the 5 winning numbers ?

    Thank you.
    Alternatively,

    there are \binom{5}{4} ways to choose 4 of the 5 winning numbers.

    The selection of 5 must contain one of the 40 remaining numbers,

    hence there are \binom{5}{4}\binom{40}{1} ways to match 4.

    There are \binom{45}{5} ways to choose 5 from 45

    so the probability of matching 4 of the 5 winning numbers with a single selection is

    \frac{\binom{5}{4}\binom{40}{1}}{\binom{45}{5}}
    Last edited by Archie Meade; May 8th 2010 at 11:38 AM. Reason: typo
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