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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?

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Part of the problem statement is missing, it seems. –  Hagen von Eitzen Jan 22 '13 at 15:55
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Probably he means: If you partition a group of $n$ elements into two lists and if $0<a<0.5$ is given, what is the probability that the smaller list has more than $an$ elements. –  N. S. Jan 22 '13 at 16:00
    
The identity of the list elements doesn't matter? If $n_1 \ge a$ with no other restrictions, you are asking how many $k$ of $\lceil a n \rceil \le k \le n$ are even. That is rather easy, but be careful with the parity of the end point of the range. –  vonbrand Jan 22 '13 at 16:28

1 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?

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