Sortostat/Optimal sorting cutoffs

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 perseve a constant dilution rate?

Solution

Definition of variables

[math]\displaystyle{ E[X_N] = }[/math] expected value of the optimal percentage that can be returned from N trials

[math]\displaystyle{ S_i = }[/math] random variable representing the percentile returned from a single trial

• all trials are assumed to be independent therefore [math]\displaystyle{ S_i = S }[/math], for all i

[math]\displaystyle{ C_i = }[/math] the cut-off percentile for the trial i positions from the final trial.

General

[math]\displaystyle{ E[X_N] = P(S\gt C_1) E[S|S\gt C_1] + }[/math]

[math]\displaystyle{ (1-P(S\gt C_1))(P(S\gt C_2)E[S|S\gt C_2]) + }[/math]

[math]\displaystyle{ (1-P(S\gt C_1))(1-P(S\gt C_2))(P(S\gt C_3)E[S|S\gt C_2]) + }[/math]

[math]\displaystyle{ ... }[/math]

[math]\displaystyle{ (1-P(S\gt C_1))(1-P(S\gt C_2))...(1-P(S\gt C_{N-1})E[S_N] }[/math]

Simplified

Since (1-P(S>C_1)) can be factored out of every term after the first above, the solution can be simplified and solved recursively.

[math]\displaystyle{ E[X_N] = P(S\gt C_N) E[S|S\gt C_N] + (1-P(S\gt C_N))E[X_{N-1}] }[/math]

base case [math]\displaystyle{ E[X_1] = P(S)*S }[/math]