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Math Help - probability

  1. #1
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    probability

    Hi can anybody help with this probability question?
    i have 8 purple socks 4 green socks and 4 red socks in a drawer. two socks are chosen without replacement. What is the probability that a pair the same colour is chosen?
    thanks
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  2. #2
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    Hello, math_lete!

    There are at least two ways to approach this problem . . .


    There are 8 purple socks 4 green socks and 4 red socks in a drawer.
    Two socks are chosen without replacement.
    What is the probability that a pair the same colour is chosen?

    \begin{array}{ccccc}P(\text{2 purple}) &=& \frac{8}{16}\cdot\frac{7}{15} &=& \frac{14}{60} \\ \\[-4mm]<br />
P(\text{2 green}) &=& \frac{4}{16}\cdot\frac{3}{15} &=& \frac{3}{60} \\ \\[-4mm] <br />
P(\text{2 red}) &=&\frac{4}{16}\cdot\frac{3}{15} &=& \frac{3}{60} \end{array}

    Therefore: . P(\text{2 purple or 2 green or 2 red}) \;=\;\frac{14}{60} + \frac{3}{60} + \frac{3}{60} \;=\;\frac{20}{30} \;=\;\frac{1}{3}


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

    There are: {16\choose2} = 120 possible pairs.

    . . There are: {8\choose2} = 28 ways to get two purple socks.

    . . There are: {4\choose2} = 6 ways to get two green socks.

    . . There are: {4\choose2} = 6 ways to get two red socks.

    Hence, there are: . 28 + 6 + 6 \:=\:40 ways to get a matching pair.


    Therefore: . P(\text{matching color}) \;=\;\frac{40}{120} \;=\;\frac{1}{3}

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