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    I have six decks of cards, all fully shuffled so the order is completely random.
    I draw two cards.
    What is the probability they are exactly the same card? (e.g 2 King of Clubs or 2 Jack of Hearts)

    Thanks
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    Anyone?
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    Sorry I also read the question wrong haha....it's jsut 5/311.
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    \frac{4}{52} * \frac{3}{51} = \frac {1}{221}

    EDIT:

    Oh, didn't read properly, six decks. Wait a while :p:

    \frac{24}{312} * \frac{23}{311} = \frac {23}{4043}

    Should be correct (I hope)

    EDIT:

    Aaah, again misread. Thought same card in meaning of ace or 4, but doesn't matter whether it's diamonds/hearts etc. Oh wells, the poster below is correct
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    (1 / 52) * ( 5 / 311) = 5 / 16172
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    I disagree with the answers so far.

    I believe the question is 'what is the probability that the second card you draw matches the first?'

    There are 6*52 cards altogether. Once the first card is drawn only five copies of that card remain.
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    5/311.

    6 decks of 52 cards per deck = 312 cards.
    you've taken one so there's 311 left.
    and only 5 of the card remain.

    so it's 0.016 or 5/311
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    The total number of (ordered) possible hands is 312\mathrm{P}2 = \frac{312!}{(312-2)!} = 97032.

    Now we restrict our potential hands to those which consist of two of the same card.

    There are 52 \times 6 ways of choosing the first card, and then only 5 ways of choosing the second card. This gives a total of  52 \times 6 \times 5 = 1560 hands which have both the same card.

    Thus the probability of drawing two cards out of the deck which are the same is \frac{52 \times 6 \times 5}{312\mathrm{P}2} = \frac{1560}{97032} = \frac{5}{311} which, of course, agrees with Gemma's answer above.
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    Thanks all you good people!

    Another one.
    Same set up, 6 decks, two cards are randomly drawn.

    What is the probability that they are the same number and the same colour? E.g King of Heart, King of Diamonds, or 7 of clubs, 7 of spades.
    Lastly, what is the probability they are the same number, but not same suit or colour?, e.g King of Diamonds, King of Clubs, or 7 of Hearts, 7 of spades?
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    As before, it doesn't matter what the first card is.

    In first case, of the 311 cards that then remain, 11 cards will count, five copies of the original, plus 6 copies of its partner (KH - KD)

    11/311

    For the last bit 6 copies of each of two suitable cards are available. (If KH drawn first, then KC or KS will do)

    So 12/311
 
 
 
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