# Does picking randomly from an imbalanced group balance the group?

If I had a bag of marbles with 75% blue and 25% red (ratio is what matters not raw number, so this applies to 100 marbles, 1000 marbles, 100000 marbles) So if I had this imbalance of 75% blue and 25% red and I started picking them out of the bag at random, would the ratios shift towards balance? Would, after a while of picking marbles at random, the ratio remain the same, 75-25, or would they become 50-50?

• Just to make it completely clear, when you talk about "balancing the groups", do you mean "among the marbles that have been selected" or "among the marbles which have not yet been selected"? (In fact this turns out not to matter, but I'm curious which is the scenario that you are imagining.) – Silverfish May 24 '16 at 13:20
• Also, are you interested in knowing what happens to the ratio on average (the simple answer is that it doesn't change), or to know from a probabilistic point of view, what are the chances of the ratio evolving in a particular way? – Silverfish May 24 '16 at 13:23
• 1. It was what was left in the bag. 2. I am now curious what the chances might be of the ratio evolving towards equal 50-50, red-blue marbles. – user116732 May 24 '16 at 15:40

Assuming you sample without replacement, then number of red balls drawn follows a hypergeometric distribution and the expected sample ratio is always 25% red marbles regardless of if you draw 5,10,20 or 100 marbles from your large bag.

The variability is in your situation a quadriatic function of your sample size, bag size and ratio of red balls. It has its maximum when n=N/2.

The implication of this is that you are more likely to sample close to 1:3 (red:blue) for small and large samples, n, regardless of your bag size N.

I hope the following R code illustrate this.

hypvar<-function(n,N,Krel=0.25) {
s2<-n*Krel*(((1-Krel)*N)/N)*((N-n)/(N-1))
return(s2)
}

var.est<-sapply(1:1000,hypvar,1000,0.25)
plot(1:1000,var.est)


N is the size of your bag, n is the number of marbles drawn and Krel is the fraction of red marbles in the bag. We assume 1000 marbles in the bag and 25% are red. The function then illustrate the variance of the number of red balls among those drawn.

https://en.wikipedia.org/wiki/Hypergeometric_distribution