First slide

Introduction to probability

Question

There are two red, two blue, two white, and certain number (greater than 0) of green socks in a drawer. If two socks are taken at random from the drawer without replacement, the probability that they are of the same color is 1/5, then the number of green socks are _________.

Moderate

Solution

Let the number of green socks be x > 0.
E: Two socks drawn are of the same color
$\begin{array}{l} \Rightarrow P (E) = P (RR or BB or WW or GG) \\ = \frac{3}{6 + x} C_{2} + \frac{^{x} C_{2}}{^{6 + x} C_{2}} \\ = \frac{6}{(x + 6) (x + 5)} + \frac{x (x - 1)}{(x + 6) (x + 5)} = \frac{1}{5} (given) \end{array}$
$\Rightarrow 5 (x^{2} - x + 6) = x^{2} + 11 x + 30$
or $4 x^{2} - 16 x = 0$
or x = 4

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