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Math Help - Conditional probability involving deck of cards.

  1. #1
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    Question Conditional probability involving deck of cards.

    I had a discussion with a friend the other day:

    Deck of cards (52 cards). 13 of them are hearts. Well shuffled, randomly dealt.

    Situation 1
    I'm sitting at a table by myself with 2 cards in my hands. Both are hearts.
    The rest of the deck is on the table.
    X = The probability that the top card of the remaining deck, when turned, is a heart.

    Situation 2
    I'm sitting at a table together with 4 other people we all have 2 cards in our hands. I do not know which cards are held by the other players. Both my cards are hearts.
    The rest of the deck is on the table.
    Y = The probability that the top card of the remaining deck, when turned, is a heart.

    Question
    X = Y ?

    And why / why not?
    Last edited by mr fantastic; October 10th 2011 at 11:16 AM. Reason: Title.
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  2. #2
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    Re: Probability problem - please help! :)

    Hello, drwmn!

    I had a discussion with a friend the other day.

    Deck of cards (52 cards). 13 of them are hearts. Well shuffled, randomly dealt.

    Situation 1
    I'm sitting at a table by myself with 2 cards in my hands; both are hearts.
    The rest of the deck is on the table.
    X = The probability that the top card of the remaining deck, when turned, is a heart.

    Situation 2
    I'm sitting at a table together with 4 other people; we all have 2 cards in our hands.
    I do not know which cards are held by the other players.
    Both my cards are hearts. .The rest of the deck is on the table.
    Y = The probability that the top card of the remaining deck, when turned, is a heart.

    Question
    X = Y ?

    And why / why not?

    The answer is YES; the probabilities are equal.


    Situtation 1

    You have two Hearts.
    The rest of the deck has 50 cards: 11 Hearts and 39 Others.
    . . P(\text{Top card }\heartsuit) \:=\:\frac{11}{50}


    Situtation 2

    You have two Hearts.
    The rest of the deck has 50 cards: 11 Hearts and 39 Others.

    Randomly select six cards from the deck
    . . and move them to the bottom of the deck.
    (These are the cards dealt to the other three people.)

    No matter what six cards were moved,
    . . the probability that the top card is a Heart remains the same.

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