# Sortostat/Optimal sorting cutoffs

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## Contents

## Problem

What is the optimal cut-off percentile for choosing a chamber to be sorted if you have N sorts (trials) remaining until you must take the sort to preserve a constant dilution rate?

## Analytical Solution

### Definition of variables

expected value of the optimal percentage that can be returned from N trials

random variable representing the percentile returned from the ith trial

- all trials are assumed to be independent therefore , for all i

the cut-off percentile for the ith trial.

### General

### Simplified

Since

can be factored out of every term after the first above, the solution can be simplified and solved recursively:

base case:

- e.g., if you have only 1 trial then you expect to get the mean of the distribution for S.

## Simulation Solution

Since our probability skills were pretty sad, we (Alex Mallet) simulated it to confirm our analytical results. MATLAB file can be found here.