Why do recessive lethal genetic diseases continue to exist?
Firstly, sorry if this is not so much an answer as a comment, but it is too long to put in the comments section, although I'll try keep it concise...

I can see exactly what you're doing however my math differs at points, and would appreciate you looking over it.

"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"

Is it not: ... =  4 - 8p + 4p^2 + 4p - 4p^2 + p^2 = 4 - 4p + p^2 = (p-2)^2

Furthermore: "The expected number of carrying: 2*p*(1-p) + 2*p^2 = 2*p - 2*p^2 + 2*p^2 = 2p" - why for the probability of carriers in mixed couples - "2*p*(1-p)" - why have you not multiplied through by two (as with expected healthy children), given that as you said "a mixed couple will have, on average, 2 healthy and 2 carrying children"?

Is it not: 2 *2p(1-p) + 2p^2 = 4p - 4p^2  + 2p^2 = 4p - 2p^2 = 2p(-p + 2)

In this respect I got different expectations. Lastly "Dividing to get the new probability for carrying the gene..." You have done (no. healthy) / (no. carriers). Is the probability for carrying the gene not (no. carriers)/(no. carriers + no. health) - that is the no. carriers as a fraction of the entire population to give a probability?

Using my expectations this would be: [2p(-p+2)] / [2p(-p+2) + (p-2)^2] = [2p(-p+2)] / [-2p^2 + 4p + p^2 - 4p +4] = [2p(-p+2)] / [-p^2 + 4] = [2p(-p+2)] / [(-p + 2)(p + 2)] = 2p / (p+2)

Therefore, with a new probability for carrying of 2p / (p+2), for our "hypothesis" of the diminishing probability of carrying to stand, we need the condition 2p / (p+2) &lt; p to stand, solving for p:

2p &lt; p(p+2) -> 2p &lt; p^2 + 2p -> 0 &lt; p^2 -> p > 0

Since, as a probability, p must be &lt;= 1, and assuming (as we are) that carriers exist and so it must be > 0, p always satisfies this condition - ergo I would conclude that for any value of p - not just small, the probability of carrying will diminish in the next generation. Just as we "hypothesised". However a quick google of "plot y = 2x/(x+2)" would indicate to me that as p approaches 0 it's rate of "diminishing" actually decreases, and it does not get exponentially smaller as you indicated. But solving for y = 0 gives x = 0 so this model would suggest the gene is never actually fully eradicated?! Which of course confuses me, although on a macro scale it might make sense, as you can't have, for example, 0.1 children "carrying", and 99.9 "healthy", only 1 and 99 or 0 and 100.

Thanks for reading this far, any response would be really appreciated, especially given I might be miles off.

Philip.