# Probability of Majorization

### Problem

### Solution

Rearrange $a's$ in the descending, $b's$ in the ascending order.

Then the partial sums of $\{a_i\}$ lie on a concave curve going through $(0, 0)$ and $\displaystyle \left((n, \sum_i a_i\right)$ and the partial sums of $\{b_i\}$ lie on a convex curve going through $(0, 0)$ and $\displaystyle \left(n, \sum_i b_i\right).$ Thus the former is always above the latter which proves the majorization.

The probability in question appears to be $1.$

Observe that the proof is valid without the assumption that the given sequences consist of integers. The same remains true if the numbers are real.

### Acknowledgment

The above solution is by Timon Knigge.

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