I'm trying to understand, in a simplistic representation of a recessive genetic disease (e.g. Tay Sachs), why wouldn't it "disappear" over time (many generations). Please excuse the use of non-professional terminology below, I hope the point I'm trying to make will be clear.
Let's assume that, for simplicity, people marry at random and each couple has 4 children. Denote the probability of carrying the "bad" gene by p, then a couple of 2 non-carrying people will have 4 healthy children, a mixed couple will have, on average, 2 healthy and 2 carrying children, and a couple of 2 carrying people will have 1 healthy, 2 carrying, and one sick child (which we assume will not have children so we ignore it in further calculations).
Assuming large enough population and random couples and everyone gets married (for simplicity): The probability of a couple of non-carrying: (1-p)^2 The probability of a mixed couple: 2 * p * (1-p) The probability of a 2-carrying couple: p^2
The expected number of healthy children per couple: 4*(1-p)^2 + 2*2*p*(1-p) + p^2 = 4 - 8*p + 4*p^2 + 4*p - 4*p^2 + p^2= 4 - 4*p - p^2 = (2-p)^2 The expected number of carrying: 2*p*(1-p) + 2*p^2 = 2*p - 2*p^2 + 2*p^2 = 2p
Dividing to get the new probability for carrying the gene: 2p / (2-p)^2 Which for small enough p is clearly smaller than p (e.g. for p = 0.05 we get ~0.0263).
If I'm not mistaken this means that the number of those carrying the genes should become exponentially small (in the number of generations).
I realize this is a very simplistic model, marriages are not random, mutations occur etc. but I would expect the basic intuition to hold. What am I missing?