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How can everyone have unique iris and fingerprints? After a certain amount of human beings have lived on earth, wouldn't it be possible to exhaust all possible combinations?

The same principle applies to DNA. Lets say counting all the nucleotides on a strand of DNA we find that there are a total of $N$. We know there are 4 types of nucleotides, thus the total number of possible DNA sequences equals $4^N$. This is exponential, and since $N$ is immensely huge, $4^N$ will be unthinkably huge as well. However, since $4^N$ is a finite number, how can DNA be guaranteed to be unique? Doesn't that contradict the pigeon hole principle?

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    $\begingroup$ There is always a chance of identical DNA sequences, and perhaps even identical biometrics between subjects. However, the chance is simply so small that it is of no problem for forensics and the likes. $\endgroup$ – AliceD May 18 '15 at 11:30
  • $\begingroup$ I guess the assumption is wrong, that only DNA is responsible for the layout of iris and fingerprints. Even people with identical DNA (like twins) shall have different iris and fingerprints? So comparing the number of resulting combinations to the number of people would probably exhaust the lifetime of our universe. $\endgroup$ – V15I0N May 18 '15 at 20:30
  • $\begingroup$ This really does not relate to DNA, specifically. I say this because there are likely not that many allelic differences that account for pigment in the iris or patterning of fingerprints. This is a developmental question. It is the randomness of how cells migrate and the timing of gene expression that dictates. Each cell will either express from both alleles of a gene or there will be gene choice. The choice however does not need to be the same for each and every cell in a tissue. So where a cell ends up in development and whether it is expressing Maternal, Paternal, or both alleles determines $\endgroup$ – AMR Dec 17 '15 at 17:38
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The uniqueness of irises and fingerprints are, as you said, limited to the number of possible permutations of irises and fingerprints.

A similar problem exists in computer science, and is known as a hash collision. Given sufficient samples, there will always be a collision for a hash of finite size. However, the sample space is sufficiently large for iris and fingerprint analysis for this to not be a problem.

A typical iris analysis system produces 2048 bits of entropy, while a typical fingerprint analysis system produces about 82 bits. Even accounting for the birthday problem, the information space is large enough that the chance of a given false positive is sufficiently low to prevent a random person from passing fingerprint/iris authentication.

For comparison, the current world human population is 7.2 billion, or slightly under 33 bits. The risk of a hash collision occurring is small enough to be negligible.

Of course, side-channel attacks against such biometric systems are relatively simple, but that's another issue altogether.

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    $\begingroup$ @Gordon That would be somewhere around 108 billion which has a collision probability of approximately 0.1% assuming a fingerprint is used. $\endgroup$ – March Ho May 18 '15 at 12:40
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    $\begingroup$ @Gordon you should also consider that the same DNA would not make the exact same individual. There is a lot of randomness involved. Consider, for example, identical twins and their differences in character. You also have random mutations occurring during an individuals lifetime or development. In any case, you should not confuse 2 people with identical DNA with reincarnation which implies some sort of continuity or link between these individuals. $\endgroup$ – terdon May 18 '15 at 12:55
  • $\begingroup$ @terdon Unrelated to this question, but this is essentially the main plot point of Jupiter Ascending. $\endgroup$ – March Ho May 18 '15 at 16:36
  • $\begingroup$ To add to what @terdon said, imagine the probability that two people, with the same DNA and chemical makeup, go through all the exact same experiences in their lifetime. The people they meet would all have to be similar enough to provide the same exact experience. Not to mention language, technology, and other things which are part of society. If the universe is really "infinite", there is an infinite number of positions the objects of the universe could be in. If you argue for a finite universe, then maybe eventually it could happen. $\endgroup$ – DoubleDouble May 18 '15 at 18:43
  • $\begingroup$ @Gordon My point was that much, much, much longer after that, if the Universe has a finite space and amount of matter/energy, and continues indefinitely without "dying", a person with the exact genetic make up and exact same life could exist on a planet exactly like Earth again. Proof: Pigeon hole principle $\endgroup$ – DoubleDouble May 18 '15 at 21:03
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@March Ho's answer is an excellent answer based on the assumption that all of the observed phenotypic variance is due to genetic variance.

Environmental variance

The genetic variance is not the only underlying variance that can explain variance in phenotypic traits. There is probably quite a lot of phenotypic variance that is caused by the underlying environmental variance between individuals. To have a better understanding of what are the different kind of variance you may want to have a look at this post for example. The heritability is typically equal to 0.3 (in fact most studies to report measures of heritability that are not significantly different from 0.3). If this was to be the case for the trait of interest, it would mean that 70% of the observed variance in fingerprints would be due to environmental variance.

Developmental Noise

Also, when talking about minor differences in phenotypes (such as fingerprints), developmental noise (inherent stochasticity of biological processes such as the decay rate of proteins or the production rate of mRNA) might be of real importance. (Raser and O'Shea, 2013) uses fingerprints as example of phenotypic trait which variance is in part determined by developmental noise.

Relevance of an answer that cares only about genetic variance

While @March Ho's answer is pleasant and interesting, I think that it is not of great relevance to the problem given the important amount of non-genetic variance that is underlying the phenotypic variance of interest.

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  • $\begingroup$ +1, good catch! I didn't think about the correlation of the two variables. $\endgroup$ – March Ho May 19 '15 at 16:59

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