Is BACK.

My favorite mathrooms were the ones that were a little open-ended in terms of what equations and values you employ, but which still have answers that different people would converge on. Hopefully the wording here is clear enough to achieve that.

UPDATE: Assume that “go black” means to be with someone who is at least 20% black.

Some more assumptions have to be made, right? For instance, how many years between generations, how many kids people have, average lifespan, etc.

You could use real world numbers and averages for those things.

The problem with using real world stats is that you have hugely different figures (birth rate, lifespan, infant death mortality) for different geographical regions. Not to mention the theoretical issue of “Black” people are probably in a region where reproduction rates are very different, than other parts of the world.

I’m all for making this a sandbox model and giving universal figures.

Assumption 3 needs a little more specificity. Does this mean, for example, that once you go x% black, you never go <x% black?

Shit. Great point. Fixing.

I agree with Morris that assumption 3 is tricky, and I like his suggestion of strictly increasing black content, for lack of a better term. You could discretize this… choose how deeply to track race (say at a 1/16th level), and then at each generation you just need to track a ‘race profile’ for society… how many men and women fit into each of the discrete race ‘buckets’.

The other key factor seems to be how many partners women have before they have their kids, as more partners => higher chance that they ‘go black’ before having kids. To this end, here’s a link that may be helpful:

http://anepigone.blogspot.com/2009/08/for-women-too-fewer-partners-means-more.html

The first table gives both frequency data on how many women in the sample have had a given number of partners, as well as the average number of children birthed by women with the same number of partners. No doubt the results aren’t globally relevant, but hey, we’re ballparking here, and I think you can extrapolate some useful numbers to base a reasonable model on. :)

Sorry, in the first sentence ‘strictly increasing’ should’ve been ‘strictly non-decreasing’.

I’m not sure that level of detail is actually necessary. Here’s my reasoning.

For each generation, any woman that is currently 20% or more black will choose a random male partner that is also 20% or more black. You can therefore work out what percentage of the offspring are what percentage “black” without needing to know the relative sizes.

This does assume each generation is “discrete”, but that’s a simplifying assumption that seems fair. And since Zach doesn’t ask “how many years” but rather just “how long”, giving the answer in generations is also fair.

(It goes without saying I’ve got an answer based on the above, but everyone else seems to be avoiding posting an answer, so I assume there’s a spoiler thing? First one of these I’ve done).

Hi GAZZA,

Usually the phrase ‘once you go black you never go back’ refers to a phenomenon by which a woman (any race) who has slept with a black man will only sleep with black men after that. This is the way this phrase is typically used in regular conversation, and I’m assuming it’s the meaning intended by this problem as well. (To account for mixed races, a natural extension Morris has proposed might be that if a woman has been with someone who is x% black, all future partners must be at least x% black.)

That said, as far as I know there’s no ban on posting the answer you get… it just happens that the model I’m playing with in my head is a little complex and I don’t have time to program it tonight.

Question for Zach: I’ve been thinking of this in terms of women choosing who to sleep with and men just being chosen. Do men also not go back after choosing black?

OK, having checked previous “Mathrooms”, it seems posting answers is OK; I assume Zach moderates them for a while.

So I get >50% black after 3 generations of offspring.

The “Generation 0″ is as defined in the problem – 20% of the population are 100% black, 80% are 0% black. The black women, all of them choose black male partners, so 20% of the next generation will be 100% black. Of the non-black women, 20% will choose black male partners (80% * 20% = 16% will be 50% black) and the rest will choose non-black male partners (80% * 80% = 64%).

So generation 1 consists of 20% that are 100% black, 16% that are 50% black, and 64% that are 0% black. To work out what happens when these guys and gals get bizzy, we apply similar reasoning. The 100% black women will choose partners that are at least 20% black. Therefore 20/100 * 20/36 = 5/45 of their offspring will be 100% black, and 20/100 * 16/36 = 4/45 will be 75% black. Similarly, the 50% black women will also choose male partners that are at least 20% black; 16/100 * 20/36 = 20/225 of their offspring are 75% black, and 16/100 * 16/36 = 16/225 are 50% black. The 64% of women that are 0% black choose completely randomly; it turns out that their offspring are 12.8% that are 50% black, 10.24% are 25% black, and 40.96% are 0% black.

This gives us a generation where 11.1% (625/5625) are 100% black, 17.8% (1000/5625) are 75% black, 19.9% (1120/5625) are 50% black, 10.2% (576/5625) are 25% black, and 41.0% (2304/5625) are 0% black. That is about 48.8% of the population are 50%+ black, not quite enough.

It’s trivially easy to see that the next generation will be, though of course I worked it out anyway. The same sort of reasoning applies. The third generation of offspring turns out to be about 2.1% are 100% black, 6.7% are 87.5% black, 12.8% are 75% black, 15.8% are 62.5% black, 17.4% are 50% black, 14.2% are 37.5% black, 9.9% are 25% black, 4.2% are 12.5% black, and 16.8% are 0% black. This is about 54.9% that are 50%+ black (6410750625/11675390625 to be precise).

The assumptions here are that each generation is discrete and that there are equal numbers of each gender regardless of “black percentage”.

But the question asks how long it will be until everyone is at least 50% black.

Your condition that the black women only choose black partners is not justified by the conditions of the problem.

That’s not what Zach meant by “never go back”, and there’s nothing in the problem statement to suggest that the formula should be gender-assymmetric.

“The black women, all of them choose black male partners”

THAT’S RACIST!

Also, I think that violates point #2 about being random.

Remember, not all members of a population are strictly heterosexual. Sexuality is a spectrum, and if folks are mating randomly, we’ll likely end up with quite a bit of bisexuality/homosexuality that slows down the mating process a bit.

Or maybe it doesn’t. After all, if mating occurs at a fairly high rate, we’ll end up with loads of pregnant females in a short period of time.

Fortunately, we can safely assume gender distributions of ~50% each. It’s probably a good idea to assume all members of the initial generation are fertile to begin with, but this may become a concern when considering Generation 0 further down the line.

Wait, wait, wait, your update just made it more complicated.

If to “go black” means to mate only with those who are at least 20% black, then it follows that someone who has “gone black” would subsequently forgo mating with a partner who is less than 20% black. Yes?

I’m not certain how mating in this problem works. That is, if a person in Generation 0 mates with a random person in Generation 0, will they mate again? Not “going back” typically refers to sexual partners of an individual. So I’m thinking that for each individual, there is a string 0, 0, 0, 1, 1, 1, … where the digits in the sequence refer lovers: 0 refers to non-black and the 1′s refer to black.

After each person has four children (assuming the gender spilt is 50/50 and no homosexuality), half of the parents+offspring will be black.

My interpretation of the problem is that women chose a number of (male) partners randomly, with the only constraint that “once you go black, you never go back.” After some number of partners they then decide to have kids. The cutoff to “go black” is at 20%, so the population will shift towards blackness until everyone is >=20% black, but will then stop shifting, so everyone being >=50% will not happen.

Using 3 partners per women, with only the 3rd one having their kids, 99.9% of people in the 4th generation are >=20% black (assuming the .8 white and .2 black is the zeroth generation).

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