Question

A bag contains n white and n red balls. Pairs of balls are drawn without replacement until the bag is empty. Show that the probability that each pair consists of one white and one red ball is $\frac{2^n}{{}^{2n}{C}_n}$.

Answer 1

Total no. of ways of selecting $n$ pairs $=C_n^{2n}$

No. of ways of selecting $n$ pairs each with one white and one red

$=2\times 2\times 2\times 2\times \mathrm{2......}n$ times

$=2^n$

$\Rightarrow$ Probability $\frac{2^n}{{}^{2n}{C}_n}$.