How is this different from Hezekiah’s concern that he would have a wicked son?
“In those days Hezekiah became mortally ill, and Isaiah son of Amoz the prophet came to him and said to him: Thus says the Lord … Set your house in order, for you shall die and not live… What is the meaning of ‘you shall die and not live’? You shall die in this world and not live in the World to Come. He said to him: Why all this? He said to him: Because you did not engage in procreation. He said to him: Because I saw through the holy spirit that children unworthy would come from me. He said to him: What have you to do with the secrets of the Merciful One? What you were commanded, you ought to have done, and what is pleasing before the Holy One, blessed be He, let Him do.”

Because wickedness and righteousness depend on free choice, but a sick child is the result of the laws of nature.

If he saw through the holy spirit that the child would turn out wicked, then his choice was already predetermined. Even if you say that there was still some chance he would choose otherwise, then here too there is some chance that the child will be healthy.

The difference is that the “chance” in Hezekiah’s case depended on choice, whereas the “chance” here is random. If you treat a possibility that depends on choice as a real probability, then don’t have children at all. Every one of us has some chance that a child will turn out one way or the other.
It is like the obligation to save wicked people even if that requires desecrating the Sabbath, although the reasoning given is so that they may keep many Sabbaths. The accepted explanation is that we must place before them the possibility of keeping Sabbaths; what they decide to do with it is their business.

From the Gemara’s wording, “and what is pleasing before the Holy One, blessed be He, let Him do,” it sounds like the Holy One, blessed be He, is the one who causes the child to be wicked. So there is a difference between saving an ordinary wicked person who chose wickedness on his own, since an ordinary wicked person can change, and Hezekiah’s child, about whom it was decreed in advance that he would be wicked—or at least that the likelihood of Hezekiah’s child changing was much lower than that of an ordinary wicked person, about whom nothing was decreed in advance.

Beyond that, one should remember that in the past the infant mortality rate was very high (I think not far from 25%, if not more). And nevertheless, people did not refrain from procreation. It reminds me of the story of Amram’s wife, who said to him: Pharaoh decreed only against the males, and you are decreeing against both the males and the females. In other words, there is value in fulfilling procreation even with a 50% chance of death.

What is the reasoning for making the fulfillment of procreation depend on a reasonable chance that the child will be healthy?
And what are the halakhic sources for this?
Is the reasoning that one is not obligated to fulfill such a commandment when the cost is so high?
Or because such a child perhaps will not go on to have descendants?

Oren, that is not how I understood it. Where do we ever find that the Holy One, blessed be He, makes a person righteous or wicked? In my opinion the meaning is: if the Holy One, blessed be He, wishes to kill him if he turns out wicked, let Him do so. I am supposed to do what I was commanded.
In Pharaoh’s case, it was a decree on the whole community, and if the whole community had refrained from procreation, the Jewish people would not exist. Here we are dealing with a single case. They are not comparable. Something like this appears as well—just as an illustration—in the Ran on Megillah: in a case of doubt whether a city was walled, we are not lenient for two days even though it is a rabbinic-level doubt, because then the commandment would not be fulfilled at all.
As for what happened in the past, again I say that if this applies to the public as a whole, then children should be brought into the world. Besides, social norms obviously depend on the era. In the past that was the likelihood of death, and so it was normal. Today the situation is different. [Not that this is the main point, but I also think the grief over the death of an infant today is different from what it was when such a death was considered natural.]

Aharon, both reasons are correct. But let us focus on the first. Because of the suffering involved, a doubtful procreation obligation certainly does not override definite suffering and anguish for the child and for them. After all, the child is not commanded, so why should I fulfill a commandment at the expense of his suffering? Do we say to a person: gain merit while your fellow suffers?! Furthermore, at the end of the chapter Hezkat HaBatim (Bava Batra 60b) we find that even when decrees are issued against Israel not to fulfill commandments, the commandment of procreation lapses, and likewise in years of famine and the like. All of these are only matters of reasoning, and the reasoning here is no weaker than that.
Beyond that, several other considerations join in here: a. They already have four sons. Although according to Jewish law they have not fulfilled the obligation of procreation, there is a view that even according to Beit Hillel the meaning is children as well—a son and a daughter are not specifically required. b. Even if they continue having children, it is not certain that a daughter will be born. c. In our times in general, I am doubtful how much one should have children at all because of overpopulation on the globe. True, for questions I am asked initially I do not permit this, but when it joins these other considerations it is another factor for leniency.

Regarding your reply to Aharon, you wrote that a doubtful procreation obligation certainly does not override definite suffering, but in the case at hand there is no definite suffering, only possible suffering. And not only that—the chance of fulfilling procreation is 50%, which is double the chance of suffering, which is 25%.
Regarding years of famine, I saw in the halakhic decisors that the whole prohibition against marital relations is relevant only to one who has already fulfilled procreation.
Beyond that, the expected value of childbirth in terms of suffering versus pleasure is overall positive (or at least not very negative), because the equation should also include the positive weight of the pleasure of a healthy child in the other 75%, which offsets the possible suffering.

In addition, there is the well-known dispute between Beit Hillel and Beit Shammai whether it would have been better or not better for a person not to have been created, and in the end they counted and concluded that it would have been better not to have been created. So how can a person fulfill the commandment of procreation at the expense of his child’s suffering—or lack of comfort?

By the way, at the beginning you mentioned that the reasoning that “one who fulfills a commandment will know no evil” is not in place in this case. In your view, is there a case where that reasoning is in place?

And one more point: nowadays, I think it is possible to do in vitro fertilization, in which an egg is fertilized outside the womb, and after a few days genetic tests can be done on the embryo to see whether it carries the problematic gene. If so, it can be left frozen and not implanted in the womb, and if not, it can be returned to the womb. Seemingly this completely solves the concern about the disease, and the obligation of procreation returns.

Oren—this is exactly why I wrote in my question that the specific gene has not been identified, to explain why the in vitro fertilization solution does not exist here.

Oren,

Indeed, it is only possible suffering here (I misspoke). But it is not measured only by the probabilities; it is also measured by the severity. This is suffering for an entire lifetime, and even after our death the child remains in the world in that state. (See Ketubot 33b, that prolonged lashes hurt even more than death.) The expected suffering cannot be compared to the expected value of the commandment. That is why I also included the other reasons.
And even regarding uncertainty in the face of certainty, I have another thought. We still need to think where we would stop. After all, they have already had 4 sons and still no daughter was born. According to your argument, they should always continue, because the expected suffering is lower than the expected value of fulfilling the commandment. But then they will almost certainly reach suffering with probability 1, because at no stage would stopping be justified.

As for years of famine, the straightforward meaning is that it is forbidden in any case. In the Shulchan Arukh, Orach Chayim 240:12, it says “if he has no sons” (the commentators wrote that it means a son and a daughter, but that is not what is written there). Simply speaking, if he already has sons it is forbidden, presumably for my reason as well: even in that there is some aspect of procreation, and I mentioned that there are views that he has fully fulfilled it. And indeed Tosafot on Taanit 11 wrote that there is a value in refraining from marital relations, except that if one has not fulfilled procreation then there is no obligation to refrain. Therefore Joseph practiced a pious stringency and refrained even though he did not have sons—look there carefully. What is the basis of the permission in such a case? Reasoning.
I also mentioned the Gemara at the end of the third chapter of Bava Batra, and there there are straightforwardly no such limitations.

The pleasure involved in childbirth does not enter the equation, because halakhically one does this not for pleasure but for the commandment. The suffering is an exempting consideration. Especially since they already have other children.

That itself is the Torah’s novelty: in ordinary cases, that suffering is not taken into account. From the very fact that it commanded procreation, we see that it does not take ordinary suffering into account. But that is when we are speaking of the suffering of normal life, not a case where there is a substantial concern of suffering and a life that is no life for the child and for his parents.

As you know, I do not think there is, in reality, any protection for those who perform commandments—at least in our times. As far as I am concerned, what is written means that one who fulfills a commandment should not let fear of harm stop him, not because harm will not happen to him. It is similar to what a friend of mine used to say: “If your head hurts, study Torah.” Not because studying Torah will heal him, but because a headache does not exempt one from Torah study—even someone with a headache should study Torah. But that is only where suffering and pain truly do not exempt one from the commandment.

If IVF can be done, then perhaps there is room for that. But I am not sure there is an obligation to do so within the framework of procreation. It may be an effort beyond what is reasonable.