Probability : Dividing a list into 2 classes

Question

I have a list of integer numbers ($n$). I am dividing it into two parts $n_1$ (smaller) and $n_2$ (bigger) such that the length of $n_1 \ge a*n$; $a$ is positive and $a \lt 0.5$. What is the probability of this event?

Answer 1

Hint There are exactly $2^{n-1}$ ways to divide a group into a smaller and a larger one.

Now if $m$ is the smallest integer such that $m \geq a*n$, you can have $n_1 \leq m \leq \frac{n}{2}$.

In how many ways can you have the smallest group consisting of exactly $m$ elements?

And be careful, when $n$ is even you might have some small problems. In that case it is not clear if $n_1=\frac{n}{2}=n_2$ works, is smaller/larger strict or not?