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Some Awful Python Code
Here’s the modeling code from the substack post.
from math import comb
from math import log10
import sys
def tprb(n, k, t):
totalp = 1
for i in range(0,k,1):
p = 1 - (t / (n - i))
totalp = totalp * p
return 1-totalp
ctoffset = 45
ctslope = -3.36
def predictCT(fragments,ampregion,fragmentsize,genomesize):
totalfragments = genomesize-fragmentsize
ontargetfragments = fragmentsize-ampregion
targetfragments = fragments*(ontargetfragments/totalfragments)
ct = ctoffset+log10(targetfragments)*ctslope
return ct
genomesize = 30000
totalng = 10
qpcrampregion = 90
basesperng = 1771000000000
rrnafraction = 0.6
fragmentsize = 1000
starttargetfraction = 0.000000001
targetfraction = starttargetfraction
totalfragments = (basesperng*totalng)/fragmentsize
reads = 100000
targetfragments = totalfragments*targetfraction
totalfragments_norRNA = ((totalfragments-targetfragments)*(1-rrnafraction)) + targetfragments
# Vary reads count
#print ("total fragments, reads, target fragments, probability of >1 read, probability of >1 read (no rRNA), predicted CT")
#for creads in range(reads,100000000,1000000):
# print (totalfragments, " ", creads, " ", targetfragments, " ", tprb(totalfragments,creads,targetfragments), " ", tprb(totalfragments_norRNA,creads,targetfragments),predictCT(targetfragments,qpcrampregion,fragmentsize,genomesize))
# Vary target fraction
#print ("target fraction, total fragments, reads, target fragments, probability of >1 read, probability of >1 read (no rRNA), predicted CT")
#while targetfraction < 1:
# targetfragments = totalfragments*targetfraction
# print (targetfraction, " ", totalfragments, " ", reads, " ", targetfragments, " ", tprb(totalfragments,reads,targetfragments), " ", tprb(totalfragments_norRNA,reads,targetfragments),predictCT(targetfragments,qpcrampregion,fragmentsize,genomesize))
# targetfraction *= 10
# Vary both
#print(", ", end='')
#while targetfraction < 1:
# targetfragments = totalfragments*targetfraction
# print(targetfraction, ", ", end='')
# targetfraction *= 2
#print()
#for creads in range(reads,5000000,100000):
# print(creads, ", ", end='')
# targetfraction = starttargetfraction
# while targetfraction < 1:
# targetfragments = totalfragments*targetfraction
# print (tprb(totalfragments,creads,targetfragments), ", ", end='')
#
# targetfraction *= 2
# print()
# Vary both - CT
print(", ", end='')
while targetfraction < 1:
targetfragments = totalfragments*targetfraction
print("{:.2f}".format(predictCT(targetfragments,qpcrampregion,fragmentsize,genomesize)), ", ", end='')
targetfraction *= 2
print()
for creads in range(reads,5000000,100000):
print(creads, ", ", end='')
targetfraction = starttargetfraction
while targetfraction < 1:
targetfragments = totalfragments*targetfraction
print (tprb(totalfragments,creads,targetfragments), ", ", end='')
targetfraction *= 2
print()
N choose K, with T targets. Probability of choosing at least one target
Looking at n choose k, but you want to get at least one of a target t. For example you have a bag of blue marbles, which contains some number of red marbles. N is the total number of marbles in the bag. K is the number of marbles being chosen. T is the number of red marbles.
This is probably well known, but I worked through it a few different ways anyway…
First a simulation (C++):
#include <iostream>
#include <vector>
using namespace std;
int main(int argc,char **argv) {
int n = 3000;
int k = 50;
int t = 30; // targets
vector<int> items(n,0);
for(int i=0;i<t;i++) {
int itm=rand()%items.size();
for(;items[itm] !=0;) itm = rand()%items.size();
items[itm] = 1;
//items[i] = 1;
}
//cout << "items: ";
//for(int i=0;i<items.size();i++) cout << items[i] << " ";
//cout << endl;
vector<int> count(t,0);
// rounds of choosing
int rounds=100000;
for(int j=0;j<rounds;j++) {
cout << "round: " << j << endl;
int tcount=0;
vector<int> selected;
for(int i=0;i<k;i++) {
int itm = rand()%items.size();
for(;std::count(selected.begin(), selected.end(), itm);) itm = rand()%items.size();
selected.push_back(itm);
//cout << "selected: " << itm << " " << items[itm] << endl;
if(i%100000==0) cout << i << endl;
}
//count items on target
for(int i=0;i<selected.size();i++) {
if(items[selected[i]] == 1) tcount++;
}
cout << "tcount: " << tcount << endl;
count[tcount]++;
}
int morezero=0;
for(int i=0;i<t;i++) {
if((count[i] > 0) || (i==0)) cout << i << " " << count[i] << endl;
if(i>=1) morezero+=count[i];
}
cout << "One or more: " << morezero << endl;
cout << "One or more fraction: " << (double)morezero/rounds << endl;
}Next I calculated this by looking at the number of combinations calculating the fraction of combinations containing at least one target and total number of combinations:
from math import comb
import sys
def totalcomb(n, k, t):
return comb(n,k)
def targetcomb(n, k, t):
total = 0
total += comb(t,k)
for i in range(k-1,0,-1): #(i=k-1;i>0;i--)
# print(t," ",i, ", ", n, " ", k-i)
# Ways of choosing i items out of t. Multiply by
total += comb(t,i)*comb(n-t,k-i)
return total
n = int(sys.argv[1])
k = int(sys.argv[2])
t = int(sys.argv[3])
print (targetcomb(n,k,t)/totalcomb(n,k,t))
print (targetcomb(n,k,t))
print (totalcomb(n,k,t))Then in terms of probability, using the probability that each draw does not contain a target:
from math import comb
import sys
def tprb(n, k, t):
totalp = 1
for i in range(0,k,1):
p = 1 - (t / (n - i))
totalp = totalp * p
return 1-totalp
n = int(sys.argv[1])
k = int(sys.argv[2])
t = int(sys.argv[3])
print (tprb(n,k,t))All approaches give the same answer, the last is the least computationally taxing…
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