First slide

Theorems of probability

Question

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

Moderate

Solution

Let the number of green socks are x > 0
E : two socks drawn are of the same color
$\begin{array}{l} \Rightarrow P (E) = P (R R or B B or W W or G G) \\ = \frac{3}{{}^{6 + x}C_{2}} + \frac{{}^{x}C_{2}}{{}^{6 + x}C_{2}} \\ \begin{array}{ll} = \frac{6}{(x + 6) (x + 5)} + \frac{x (x - 1)}{(x + 6) (x + 5)} \\ = \frac{1}{5} (given) \\ \Rightarrow & 5 (x^{2} - x + 6) = x^{2} + 11 x + 30 \\ \Rightarrow & 4 x^{2} - 16 x = 0 \\ \Rightarrow x = 4 \end{array} \end{array}$

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