# How many humans would there need to be for almost every body to have a doppelganger?

There are a number of stories about persons with exact look alikes. In general this seems to be a rather rare phenomenon. But if you assume more and more humans it should become also more likely that somewhere out there is a doppelganger. My personal rationale is that all humans come from the same ancestors there is a somehow limited number of genes that can be combined - at some point there will need to be duplicates.

So the question is: are there any numbers principles that would help estimate how many humans it would need for more people having a doppelganger than not?

To specify doppelganger I would consider a look alike person, same height, similar age, same gender, hair colour, eye colour, skin colour. They should look and sound very similar - like identical twins. Although I would ignore, haircut, tatoos, scars and any other cosmetic alteration. Especially I don't care if they are genetically identical - looks and sounds alone is enough. For this question I want to ignore cloning and twins. Also lets assume between the distant human groups there is no evolution or mutation that would separate them into different ethnics. Also no "mirror worlds".

Just by pure luck this story was featured today. Does anyone understand the maths well enough whether this applies to my question? Are we already enough humans?

update 1

After the first answers pouring in I see I should clarify some points. I am looking for cases that would be good lookalikes. As there are already a few around - a guess would be at least 100 (looakalikes so spooky close that I would classify them as doppelganger) for the 10 billion people we are. The answers based purely on genetics seem way to conservative as it would be near impossible to have a single doppelganger pair. I think it will be necessary to adjust for.

From mathematics the point to the birthday paradox is very valuable. But the birthday paradox calculates how likely it is for at least one pair to share a birthday. The number i am looking for is the probability for at least half of the people share the "birthday" (look in this case). I had a go at the equation - but I personally I think this problem needs to be tackled very different and is not trivial enough for me.

I know I'm new here so let me know if you think these clarifications "break" the question and should be stated as new one.

• You're going to have a serious demarcation problem here. Most of the features you're considering occur on continuums and you'll have to pick an arbitrary resolution. Are 170.1 cm and 170.9 cm different heights? Are auburn, red, and brunette 3 hair colors, 2, or 20? These are actually the easier cases. When you're talking about enumerating facial structures, that's a headache. – SudoSedWinifred Sep 26 '16 at 6:51
• Good point. I am aiming for what most human would be able to discriminate casually. I recall seeing a paper that not all facial features are equally important. – bdecaf Sep 26 '16 at 7:17
• one thing to point out, even identical twins can be identified as separate people by the time they get to adulthood – Chris J Sep 26 '16 at 15:18
• Why is the answer not 2? – xDaizu Sep 27 '16 at 16:28
• @xDaizu: While 2 people could be dopplegangers, it's statistically quite unlikely. The idea is that if you had a trillion quintrillion bazillion people, the odds are very high that at least one other person looks (sufficiently) like any particular person you choose. The question is how much lower could you still say the same thing? – MichaelS Sep 28 '16 at 6:14

It should be thought of as virtually impossible, because the number is going to have to be so large that random mutations are going to start to play a large factor.

A study was done in 2014, Morphological and population genomic evidence that human faces have evolved to signal individual identity, on this topic. It appears there was a tremendous evolutionary advantage to varied identifying features, so the amount of diversity is very large. From the study, they found 58 regions of DNA defining facial structure and 356 regions which affected height. The journal article was interested in real differentiation between individuals, not hypothetical potentials, so they do not have a number of variants of each region. That number would have been required in order to fully answer this question. However, we can attain a lower bound by assuming each region has a mere 2 genotypes. In this case, the number of facial structures alone would be 2^56, which would be 72,057,594,037,927,936 faces. That's 72 quadrillion faces, using a tremendously conservative assumption! With that many people, we should also start expecting to see random mutations playing a part, and it is reasonable to expect at least one individual will have a unique face simply due to genetic roulette.

The other thing to consider is that the definition of a "doppelganger" would be culturally dependent. Studies have shown that, while Asian individuals can easily uniquely identify other Asian faces, those of European decent have much more trouble doing so. Presumably this is a learned behavior, as those in Asian communities spend more time around Asian faces. However, this effect is pronounced enough that it is common in some areas to hire Asian bouncers for clubs, simply because they do a better job of identifying the faces of Asian troublemakers. So to come across a real number to answer your question, this complexity would have to be nailed down.

• That is a good find. But something bothers me. Even if i would consider a meager 75% - say 42 features - it gives a number way larger than people alive. Meaning it would be extremely unlikely to have any look-alikes at all - but it seems there are quite a few around. – bdecaf Sep 26 '16 at 21:40
• @bdecaf The question was originally asking how many people it would take for everyone to have a doppleganger (which has been changed since my answer). It takes quite a lot fewer individuals when you just need to have a high probability of finding a single doppleganger. This is known as the Birthday Paradox. This also does not address the more complicated issue of whether or not a particular individual identifies two people as dopplegangers. The question is tagged hard-science, so I limited myself to genetics I could quote from a journal. – Cort Ammon Sep 26 '16 at 22:10
• No i didn't change the question. Feel free to review the change log. – bdecaf Sep 26 '16 at 22:16
• @bdecaf Ahh, you're right. Guess I sped read too fast (I saw that it was edited, and made assumptions). The number is still going to be very big if you're interested in it being more likely than not to have a doppleganger at a genetic level. We'd need more content regarding what we consider to be a doppleganger before a final number could be reached. There's also questions about things people might do to increase their uniqueness, such as scarring and tattos. – Cort Ammon Sep 26 '16 at 22:29
• @bdecaf almost all, it like 99-90% so picture changes not so much. Also read it like all have to have doppelgangers. Picture changes when you ok to have few percents or less doppelgangers. – MolbOrg Sep 27 '16 at 1:39

Basic combinations equations would probably give you a ball park figure:
n genes in each chromosome pair
in combinations of r genes
Total possible combinations for a given pair = nCr = n!/(r!(n-r)!)
Repeat for each chromosome pair
Subtract impossible options, e.g., YY pairs in the 23rd chromosome pair
This gives you a ball park of how many possible unique individuals there are.

Now, group them based on an arbitrary rule, e.g., 5% difference or less between 2 individuals means they're identical, so how many distinct 5% slots can you make?

Let's say, we need N nucleotides to match for two people to be dopplegangers. Also, let's say the nucleotides are random, so the probability of these N nucleotides to match in two random people - making them doppleggangers - is (1/4^N). The math expectation for a person to have a doppleganger in a population of M people will be M/4^N.
You want at least half of the population to have dopplegangers, so let's say the expectation is 1/2 (*), M=4^N/2.

What that means:
M = 8 billion, N = 17 - half of Earth population would have a doppleganger of random age if the appearance is encoded by 17 nucleotides.
M = 8 billion / 64, N = 14 - half of Earth population would have a doppleganger of the same age if it's encoded by 14 nucleotides.
M = 10^80, N = 133 - if we have as many people as there are atoms in the currently observable universe, you have 133 nucleotides to encode the appearance. Just for the scale, human genome has 3 billion nucleotides.

(*) Yes, in this case some people would have several dopplegangers and more than a half would have none, at the end the constant doesn't matter that much anyway.

• I fear the simplifications you make don't apply here. The exact expectation for a person is 1-(1-p)^N. the formula you use is an approximation for small p and n. But here we need 50% where it no longer holds (also it could give probabilities beyond 100%). The other point is that it doesn't matter which of the humans have dopplegangers which should lead to a much smaller number. – bdecaf Sep 28 '16 at 6:47
• @bdecaf, "the exact expectation for a person is 1-(1-p)^N" - why? Do I understand correctly, (1-p) is a probability that one nucleotide mismatches? Then (1-p)^N is a probability all N nucleotides mismatch, and 1-(1-p)^N is a probability at least one nucleotide match. But we need the probability that all N nucleotides match, which is p^N. – user8808 Sep 28 '16 at 10:56
• I made some typos writing it on the phone. p is supposed to be the probability two humans are 'similar enough'. N the number of humans. Sorry about that. To write it out exactly as you did P_{has at least one doppelganger} = 1 - (1 -1/4^N)^M. The simplification of it  ~ M * 1/4^N only works as long as the number is much smaller than 1. – bdecaf Sep 28 '16 at 15:07
• But the more I think about it you could be right that it might turn out that the number I am looking for is when the probability hits 0.5 and all the birthday paradox thinking was overkill. – bdecaf Sep 29 '16 at 8:55

I don't think that this question is answerable - simply because it's too hard to define what counts as the same and what doesn't.

If the standard required is just 'pass physically around casual acquaintances' then I'd imagine that we're already there - I struggle to believe that out of a million, ten million or at most 100 million that there isn't someone who can pass for me. The best evidence that I can come up with this is celebrity lookalikes - any very famous person will almost certainly have a decent lookalike (and probably in the same country) which would imply that - given enough incentive - that you could find a pretty good double for anyone (and therefore everyone).

At the other end of the scale, to convince someone who knows that person intimately (family member, lover, best friend) would seem almost impossible - as someone's posted above, there are quadrillions of possible faces; and that's before you consider height, build, etc.

So at best, we can say that whatever the population of your world, there is a doppelganger for everyone - but it's hard to say how close a match they would be.

Your problem is not comparable to the birthday paradox. There is 365 dates of birth possible. If you increase the nomber of people it is more likely to have a pair with the same birthday since the number of date stay the same.

There is no limitation in the appearance of people so as you increase the number of people you increase the number of appearance possible. The ratio in the population of doppleganger will probably stay the same.

• Hi, this question uses the assumption that there is a limited number of faces to go around, as they still need to be human, and humans just have a finite number of genes going around. This number may be huge - but the aim is to explore how many humans there would need to be. – bdecaf Sep 28 '16 at 15:16
• So their is an inquisition around ? Purifying the race and burning people or something like that ? DEATH TO THE MUTANTs ! – Rigop Sep 28 '16 at 15:49