Calculating the Expectancy for Fulfilling a Mitzvah

Probability expectation. My question is whether, from a mathematical perspective, the Gemara's preference for one of the two options is justified.

I don't understand the question. The option in the Mishnah allows the problem to be solved completely. In contrast, in the option in the Gemara it is clear that some of the yibbums were not made on the correct yibbum. What does this have to do with the difference in probabilities? Probability for what?

There are two extreme options:
A. That one of the brothers will marry each of the wives and the other brothers will marry each of the wives, so that *only* one of the brothers (the one who marries) will *definitely* fulfill the commandment of marrying (with whichever of the wives he marries).
B. As described in the Mishnah, so that *it is possible* that *each* of the brothers will fulfill the commandment of marrying.

So that the expectation of fulfilling the commandment of marrying can be calculated for each of the options, it will become clear whether the Mishnah's preference for the second option is justified from a probabilistic perspective.

I explained above why this is not correct, and I will come back to it again. The Gemara clearly states its consideration for preferring the Mishnah's way, and it has nothing to do with probabilities. In the Mishnah's way, it is possible that everyone will marry the right woman (and it may not), without any connection to the probability of that happening. In the Gemara's way, this is not possible, and therefore it is rejected. We are not comparing the expectations of a number of marriages here, which is correct in one of the ways. This is similar to what the Gemara in the Rish B'M says: "The division can be true."

Still, I was interested in knowing what the expected value of each of the options is. I would appreciate it if you could write the calculation.

If we follow the Mishnah's path, then each brother chooses a different woman. Now we need to decide what the question is: how many on average will choose the right woman or what is the chance that everyone will choose correctly. These are different questions. The second question is irrelevant to the Gemara's suggestion because there the chance of that is 0.
Regarding the first question, in the Gemara's outline the answer is of course 1. Regarding the Mishnah, the answer is not simple (it is a distribution of Shabbat points). If you are interested, you can see here:
https://he.wikipedia.org/wiki/%D7%91%D7%9C%D7%91%D7%95%D7%9C_(%D7%A7%D7%95%D7%9E%D7%91%D7%99%D7%A0%D7%98%D7%95%D7%A8%D7%99%D7%A7%D7%94)

Actually, regarding the duration of the vulnerability, the answer is simple: 1.
See an explanation here: <a href="https://math.stackexchange.com/questions/783623/fixed-points-in-random-permutation&quot; rel="nofollow">https://math.stackexchange.com/questions/783623/fixed-points-in-random-permutation</a&gt;
Note that this is also the result of the Gemara's proposal. This of course strengthens the Gemara's claim that these two options are equivalent, and the Mishnah's preference is only because it can completely exist.

Continued from the email:
Following the discussion on the website, the graph is attached.
The expected number of yibbums in both options is 1. Do you have an understanding (beyond the Gemara reasoning) of what underlies Chazal's preference for the provided option?

This is the graph:

https://drive.google.com/file/d/1EMdf2evwylsp4JNLMpFansrosyqag27a/view?usp=sharing

I didn't look at the details, but why does it matter? The Gemara explained its preference well. Beyond that, the graph here is not the relevant graph, but its derivative (or difference graph, in the discrete case). The relevant graph is how many at least yivums there are in each situation (which is the area under this graph). For example, the chance of four yivums is 0, but not because there is no situation of four yivums but because if four scored then necessarily the fifth also scored, and therefore there is no situation of exactly four. The relevant graph is monotonically decreasing, of course.

Thanks for the comment. From Chazal's choice of the assured option over the certain option, even though both have equal expectancy, can't we conclude that they prefer risks?

I didn't understand. They chose an option that could be complete but there is no certainty there.

Perhaps you meant to say that it is better to act in a way that we achieve the full achievement even though there is a concern that none of them will be correct than the possibility that there will definitely be one correct answer but it is impossible for all of them to be correct. This certainly exists here.

A more conservative approach would be to settle for one answer for sure, than the permissive approach in which in more than 1/3 of the cases there is no answer. This means that the Sage's approach is taking risks.

To verify your comment about the graph, you claim that it is better to present the graph:

Σp(i)-Σp(i+1)
Where:
i = 0, 1, …, N-1
Where N is the number of women

This is too sweeping a conclusion.
Regarding the formula, I did not mean that. On the contrary, the chance that there will be at least 3 answers is the sum of 3, 4, and 5. In the graph p(n) describes the chance of an exact n, not the chance of at least n. This is not a relevant number for our discussion. The difference you wrote here, in my opinion, does not give anything relevant.