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Math Help - Statistics Card Problem

  1. #1
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    Statistics Card Problem

    Two playing cards are drawn from a deck, they are the 5 of Spades and the Queen of Hearts. What is the probability that the third card drawn has a value between the 5 and the Queen? (Note: suit does matter here.)
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by teekayoh View Post
    Two playing cards are drawn from a deck, they are the 5 of Spades and the Queen of Hearts. What is the probability that the third card drawn has a value between the 5 and the Queen? (Note: suit does matter here.)
    Let event A=\text{drawing a card between 5 and queen on the third draw, where suit does matter}
    Let event B=\text{2 cards are 5}\spadesuit\text{ and Q}\heartsuit

    So you are looking for this:

    P(A|B)=\frac{P(A\cap B)}{P(B)}\backepsilon P(B)\neq0

    So we see here that P(A\cap B)=P(B)P(A|B)

    It should be evident that P(B)=\frac{\binom{13}1\binom{13}1}{\binom{52}2}=.1  27 and that P(A|B)=\frac{30}{50}=\frac{3}{5}, assuming that the cards are not replaced.

    Thus, P(A\cap B)=.127\times .6=\color{red}\boxed{.076}

    Does this make sense?

    --Chris
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  3. #3
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    ~

    Hello, teekayoh!

    Two playing cards are drawn from a deck: 5\spadesuit and Q\heartsuit.
    What is the probability that the third card drawn has a value
    between the 5 and the Queen? (Note: suit does matter here.)

    . . Not sure how the suit matters.

    There are 50 cards remaining in the deck.

    There are 24 cards between 5 and Queen: four each of {6, 7, 8, 9, 10, J}

    Therefore: . P(5 < x < Q) \;=\;\frac{24}{50} \;=\;\frac{12}{25}


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    If "between" includes 5 and Q, among the 50 remaining cards,
    . . there are 24 cards mentioned above, plus \{5\clubsuit,\:5\heartsuit,\:5\diamondsuit,\:Q\clu  bsuit,\:Q\spadesuit,\: Q\diamondsuit\}

    Therefore: . P(5 \leq x \leq Q) \;=\;\frac{30}{50} \;=\;\frac{3}{5}

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