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Math Help - Married couple raffle question.

  1. #1
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    Married couple raffle question.

    Hey, I don't think I am doing this question correctly, my solutions just seem to simple.

    Question:
    There are 8 opposite-sex married couples at a party. Two people are chosen at random to win a door prize.
    a) What is the probability that the 2 people will be married to each other?
    b) What is the probability that the 2 people will be of the same sex?
    c) If 6 people are chosen, what is the probability that they are 3 married couples?

    Solution:
    a)  n(s) = 8C2 = 120
     n(married couples) = 8
    So, that means that it's 8/120, or 1/15.

    b)  (\frac {8}{16})(\frac {7}{15})
     = \frac {7}{30}

    c)  n(s) = 16C6 = 8008
    There are 8 married couples, so 8C3.
     \frac {56}{8008}
     \frac {1}{143}


    I am unsure if any of these are correct, as I am not used to doing these sorts of problems with things like married couples.
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  2. #2
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    Lexington, MA (USA)
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    Hello, Kakariki!

    The problems are simple, but require some careful thinking.
    And you did fine!


    There are 8 opposite-sex married couples at a party.
    Two people are chosen at random to win a door prize.

    a) What is the probability that the 2 people will be married to each other?

    Solution:

     n(s) \:=\: _8C_2 \:=\: 120

     n(\text{married couples}) \:=\: 8

    So, that means that it's: . \frac{8}{120} \:=\:\frac{1}{15} . . Right!


    b) What is the probability that the 2 people will be of the same sex?
    The first person can be any of the 16 people: . \frac{16}{16} \:=\:1

    The second must be one of the other 7 people of the same sex: . \frac{7}{15}

    Therefore: . P(\text{same sex}) \;=\;\frac{7}{15}

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

    Another approach . . .

    There are 120 possible outcomes.


    \text{We want: }\:\begin{Bmatrix} \text{2 men:} & _8C_2 \:=\:28\text{ ways} \\ <br /> <br />
\text{or} \\ \text{2 women: } &  _8C_2\:=\:28\text{ ways} \end{Bmatrix}

    Therefore: . P(\text{same sex}) \;=\;\frac{56}{120} \;=\;\frac{7}{15}




    c) If 6 people are chosen, what is the probability that they are 3 married couples?

    n(s) \:=\:_{16}C_6 \:=\:8008

    There are 8 married couples, so: . _8C_3 \:=\:56

    Therefore: . \frac{56}{8008} \:=\:\frac{1}{143} . . Yes!
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