A probability problem by Prithwish Roy

The number of ways for $n$ boys and $n-1$ girls to queue is the binomial coefficient $\binom{2n-1}{n}$

The number of ways for them to queue such that the given condition is satisfied is the Catalan number $C_n = \frac{1}{n+1}\binom{2n}{n}$ .

Therefore the probability of the condition being satisfies is $\frac{\tfrac{1}{n+1}\binom{2n}{n}}{\binom{2n-1}{n}} = \frac{2}{n+1}$

Setting $n=19$ yields $\frac{2}{19+1} = \boxed{0.1}$