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Math Help - Card probability question

  1. #1
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    Card probability question

    Hey everyone, a co-worker and I were playing a board game and afterward were trying to calculate the probability of drawing cards. Neither of us could come up with an answer and so I thought I'd ask here before I go mad trying to puzzle it out.

    The way it works is this. There are 29 cards in the deck. 27 of them are unique (let's pretend they're marked 1 thru 27. And the other 2 cards are marked "A". So they match each other but none of the others. We were simply trying to figure out if you were to draw 5 cards at random from the deck. What are the odds that you'll draw the 2 "A" cards?

    If anyone can help with the math on this one, my sanity would greatly appreciate it.

    Thanks in advance, Allan.
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  2. #2
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    Lexington, MA (USA)
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    Hello, Allan!

    I simplified the wording of the problem.


    There are 29 cards in the deck: 2 A's and 27 Others.

    If we draw 5 cards at random from the deck, what is the probability of getting the 2 A's?
    There are: . _{29}C_5 \:=\:{29\choose5} \:=\:\frac{29!}{5!\,24!} \:=\:118,\!755 possible hands.


    We draw 5 cards and want: two A's and three Others.
    . . There is: . {2\choose2} \:=\:1 way to get two A's.
    . . . There are: . {27\choose3} \:=\:2,\!925 ways to get three Others.
    . . . . . . Hence, there are 2,\!925 ways to get two A's.

    Therefore: . P(\text{two A's}) \;=\;\frac{2,\!925}{118,\!755} \;=\;\frac{5}{203}

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  3. #3
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    Thanks! I have a quick follow up question.

    There are 30 cards in the deck: 3 A's and 27 Others.

    If we draw 5 cards at random from the deck, what is the probability of getting any of the 2 A's?
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