I’m sure some of you will be left sadly shaking your head that I don’t know this, but I don’t…
For those unfamiliar with the game, you need to know these two things as background:
In The Sims a baby’s gender is determined at birth, not at conception like real beings. This means that you can save the game just before a sim gives birth, and if you don’t get a baby of the gender you want, exiting without saving and re-entering the game gives another shot of getting the gender you want. You can continue this way indefinitely.
The odds of having a boy or girl are 50-50. It’s rumored - not proven - that the game algorithm will try to compensate for having an overwhelming majority of one gender in a neighborhood by giving you more babies of the minority gender, but we’ll suppose a neighborhood that doesn’t suffer this imbalance.
Suppose I want a couple to have two girls, and only two girls. Any boys born wouldn’t be saved.
How is probability effected by having them produce two singletons versus a set of twins? If I wanted two singletons I’d save after the first girl, and then continue entering and exiting the game without saving until the second was acquired. With twins, obviously, I’d have to exit without saving and reenter after each male birth(s), only save the game once I got girl twins.
My gut tries to tell me that it will take more attempts to get a set of girl-girl twins than two singleton girls because there are three possibilities with each twin birth (girl-girl, girl-boy, boy-boy) rather than just two. What am I missing that makes this wrong? Something to do with saving the game after the first singleton girl is born, I suspect, but I’m not sure how.
It depends on what process twin genders are assigned by, but on either of the two most plausible scenarios, your get is telling you something odd. Let’s just talk probabilities and ignore saving for a second, since that seems to be your question. When you produce two singletons, there aren’t two possibilities, but four equiprobable ones (on the assumptions you’ve given us): boy-then-boy, girl-then-boy, boy-then-girl, and girl-then-girl. There’s a 1 out of 4 chance of getting girl-then-girl.
With twins, we don’t know how the computer internally computers the genders: does it flip coins twice (so that it does the same thing as the singletons, but then throws away the order, making {boy, girl} have probability 1/2 while {girl, girl} and {boy, boy} have probability 1/4), or does it flip a three-sided die once, so to speak? It probably flips coins-twice, giving the same 1 out of 4 chance of two girls, but it may do the 1 out of 3 thing. In the latter case, it’s easier to get girl-girl twins than girl-then-girl singletons.
This is ignoring any complicating factors like the computer tossing in some calculations regarding identical vs. fraternal twins or all that. Basically, it all depends on what the programmers chose to do to assign twin genders; without that information, we don’t know, though we can guess what they chose.
We can tackle the problem assuming the chance of having a boy and a girl are the same. It does make a difference, just not very much for twins. Imagine you were trying to get 10 girls, not just two (virtual mothers have big wombs). The probability of having all 10 girls at once is (1/2)^10 = .00097656. One divided by that figure is the expected number of trials until it occurs: 1,024. (Technically speaking this is a geometric distribution with parameter (1/2)^10).
If instead we “saved” after each girl, we would get a girl on average every two tries. To get ten girls would take an average of 20 trials. (This is a negative binomial distribution with parameters .5 and 10).
If you just want twins, the expectations are the same: 1/(1/4) = 4 for the geometric and 2 * 1/(1/2) = 4 for the negative binomial. In the 10-girl case the difference is quite noticeable. We’re comparing 2^n with 2*n, more or less.
ETA: Although, I don’t think you are addressing the OP’s concern. She doesn’t want the difference between saving and not saving, but the difference between twins and not-twins.
Her motivating idea seems to be that when you throw saving into the mix, then what happens with singletons vs. twins is that, if you roll “girl then boy” in singletons, you can revert to the save in the middle, preserving as guaranteed the girl and re-rolling the boy. But if you roll “girl and boy” in twins, you have no choice but to revert both of them in going to an old save, and thus lose the guaranteed girl.
I.e., saving lets you preserve partial success at a more fine-grained level with singletons than with twins. Let’s make it clear that this has nothing at all to do with the fact that the ordering information available with singletons gives a different number of distinguishable result configurations than is available with twins, though (i.e., the “three possibilities” logic with twins has nothing to do with it).
Anyway, as it happens, this finer-grained saving doesn’t help at all if we look at the average number of rolls it takes to get N girls (2N = 4 (N/2) rolls in each case). Why is that? Well, the fact that generating girls by twins instead of by singletons produces girls twice as fast exactly cancels out the fact that generating girls by twins gives less opportunity for saving partial results than generating girls by singletons.
If you don’t save, then twins and not-twins are… indistinguishable … except that in the twins case the computer rolls two dice once, vs. rolling one die twice with not-saving singletons.
It’s true that the expected number of rolls is the same in the twins case, but it is not true in the case of N kids–see my previous post.
Right, except there seemed to be some confusion in the OP that the computer would probably roll one three-sided die once in the twins case.
Ah, I had misread your post again, I think. You were discussing triplets, quadruplets, and so on? So that, to get M girls using N-plets, you would on average take M/N * 2^N many rolls? Yeah, that seems to be what you were saying.
… except that in the twins case the computer rolls two dice once, vs. rolling one die twice with not-saving singletons.