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Topic: Common Surname (Read 569 times) |
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Sameer
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Common Surname
« on: Nov 29th, 2004, 10:33am » |
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Someone at work bothered me with this puzzle. Here it goes
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I figured human being will share a same surname after infinite generations. Could anyone approve or disapprove it?
Let me clearly define the population model we’re talking about.
1. In the beginning, there are limited human beings, which have limited surnames.
2. Every human can be randomly either male or female and has only one surname. Again, people randomly choose their mates.
3. One male and one female produce from 0 to any reasonable number of descendants.
4. Every descendant’s named after his/her father’s surname.
5. No one lives forever.
Note, no new surnames are introduced to the model. Human could become extinct.
We can assume the chance that a child is boy or girl is 50:50. I think it works anyway as long as it’s not 1:0. Human can mate more than once with opposite sex and people could kill people, just like the reality. Though the facts are totally irrelevant and unnecessary to math.
Given the 5 laws of the population model, I say human being will eventually share a same surname after infinite generations.
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asterix
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Here's how I would model it.
1. Ignore all girls. Just assume enough exist for boys to get married. (I guess that's why I'm single).
2. Every family will have 0 to any reasonable number of boys who will have children and pass on their name. If it's high enough to cause growth in population, then most family names will grow (a few unlucky ones will die out before they reach critical mass). If the odds favor most surnames dying out, then the same odds imply that all names will die out, given infinite time.
3. You have given no reasons we are to consider why these family names are in competition with each other, why a decrease in the numbers of one name would favor an increase in the numbers for any other name. There could be such reasons.
For example, given limited space and food, the population levels could remain constant. In that case random fluctuations would occasionally wipe out a name, and the remaining names would grow to fill the void. Eventually one name might become universal. Without getting into cultural preferences or evolution of one families genetic strength, I can't see any other reason why all surnames should not have the same destiny in infinity.
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Icarus
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Re: Common Surname
« Reply #2 on: Nov 30th, 2004, 8:46pm » |
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This something of an "Eve in reverse" situation: Geneticists believe that all people are the descendants of a single woman who lived a 1 or 2 hundred thousand years ago. This is based on mitochondrial DNA evidence. One conclusion from this is that every female line of every other female living at that time has died out.
In this case we are talking about male lines instead of female ones, but the same principles operate. Sameer's contention is that given enough generations, all but one of the current male lines will die out, just as all but one female line already has. (In fact, both have probably occured more than once.)
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John_Gaughan
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Re: Common Surname
« Reply #3 on: Dec 7th, 2004, 6:58am » |
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If I remember correctly, the evidence was mainly due to environmental factors (e.g. small scale extinction events) that wiped out entire clans. In the absence of these events (given the rules in the original post), would this still happen? I don't think so.
I think eventually, given truly random mating and baby sexes, there would only be one family name left. Clearly, if everyone having a given name has only girls, that name will die out. With truly random events, this must happen eventually. Given a truly random series of digits, for example, there will be a sequence of a million zeros in a row in there somewhere, and this will be perfectly random. The same principle applies -- at some time, we could reasonably expect an entire clan/house to die out because they spawned zero male children. Over an infinite time span, this would happen to enough clans that one or two would remain.
Of course, ALL children could be girls, killing the entire species 
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rmsgrey
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Re: Common Surname
« Reply #4 on: Dec 7th, 2004, 6:34pm » |
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My intuition is that this depends on the expected number of sons per man. If it's less than 1, then eventually extinction will happen. If it's sufficiently large, then (typically) only a few names will die out - the rest will have an infinite number of descendants each. I think those are the only two possible cases if each man's number of sons is independent - you can't have a finite non-zero expected final population because it's not stable - there's a finite non-zero chance of extinction in each generation, and if the population's bounded above, the extinction probability per generation is bounded below, so the overall extinction probability is 1.
Genetic drift (the phenomenon whereby the gene-pool of later generations is a decreasing subset of that of the first generation, eventually giving rise to complete homogeneity) relies on two things - a small enough population that the drift rate is significant compared to the mutation rate, and aproximately zero-growth - a rapidly growing population multiplies (almost) all variations, a rapidly shrinking population ends up extinct, and a large population takes too long for any given variation to die out completely, meaning that mutation will introduce variation faster than drift eliminates it.
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