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Math Help - Probability

  1. #1
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    Probability

    A and B take it in turns, starting with A, to take a card, without replacement, from a pack of 10 cards containing the numbers 1,2,2,3,3,3,4,4,4,4.

    The first player to select a "4" card is the winner.

    Find the probability that B wins the game.
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  2. #2
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    Hello, BabyMilo!

    Starting with A, players A and B take turns taking a card, without replacement,
    from a pack of 10 cards marked: 1, 2, 2, 3, 3, 3, 4, 4, 4, 4.

    The first player to select a "4" card is the winner.

    Find the probability that B wins the game.
    There are: 4 Fours and 6 Others.

    A tree diagram makes the problem clear.

    There are three scenarios in which B wins.


    [1] A draws an Other, then B draws a Four.
    . . .This probability is: . \frac{6}{10}\cdot\frac{4}{9} \:=\:\frac{4}{15}

    [2] A draws an Other, B draws an Other,
    . . . A draws an Other, then B draws a Four.
    . . .This probability is: . \frac{6}{10}\cdot\frac{5}{9}\cdot\frac{4}{8}\cdot\  frac{4}{7} \:=\:\frac{2}{21}

    [3] A draws an Other, B draws an Other,
    . . . A draws an Other, B draws an Other,
    . . . A draws an Other, then B draws a Four.
    . . .This probability is: . \frac{6}{10}\cdot\frac{5}{9}\cdot\frac{4}{8}\cdot\  frac{3}{7}\cdot\frac{2}{6}\cdot\frac{4}{5} \:=\:\frac{2}{105}


    Therefore: . P(B\text{ wins}) \;=\;\frac{4}{15} + \frac{2}{21} + \frac{2}{105} \;=\;\frac{8}{21}


    But check my work . . . please!
    .
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