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100+ Answers
Nandu Aditya

very very very unlikely.

Because in the second generation it leads to inbreeding.
Effects of Inbreeding,

The partners in these relationships that inbreed, simultaneously jeopardize their offspring's fitness and bring upon them great evolutionary risk. Inbreeding opens the door for prevalence of deleterious recessive alleles by increasing their chance for homozygosity and thereby, impeding the fitness of the offspring. With continuous inbreeding, the loss of genetic variation magnifies, and in turn, enables the expression of such recessive alleles following boundless negative implications.

In other word inbreeding leads to lack in mental and physical fitness. Specifically, inbreeding has been found to decrease fertility. Furthermore, such offspring are known to have a lower IQ and higher risk for mental retardation

Present day cheetahs are an example of this.

About 10,000 years ago, all but one species of cheetah became extinct due to climate changes. There was a drastic reduction of that one species’ numbers and the remaining ones were forced to breed, even though they were close relatives. Since then, all cheetahs are closely related.
This event caused an extreme reduction of the cheetah’s genetic diversity, known as a population bottleneck, resulting in the physical homogeneity of today’s cheetahs. Poor sperm quality, focal palatine erosion, susceptibility to the same infectious diseases, and kinked tails characteristic of the majority of the world’s cheetahs are all ramifications of the low genetic diversity within the global cheetah population.
When you look at two individual cheetahs, related cheetahs share about 99% of their genes, when in most species’ that number is about 80%. This lack of genetic diversity can cause a lot of problems, because it makes the species as a whole more susceptible to disease and less adaptable to new environments. This means that a single deadly virus could kill off all wild cheetahs in the world.Extreme inbreeding affects their reproductive success with small litter sizes and high mortality rates.

There has been a similar case of bottleneck in case of humans,


The Toba catastrophe theory as presented in the late 1990s to early 2000s suggested that a bottleneck of the human population occurred c. 70,000 years ago, proposing that the human population was reduced to perhaps 10,000 individuals,when the Toba super volcano in Indonesia erupted and triggered a major environmental change. The theory is based on geological evidence of sudden climate change and on coalescence evidence of some genes (including mitochondrial DNA, Y-chromosome and some nuclear genes)and the relatively low level of genetic variation in humans.

In 2000, a Molecular Biology and Evolution paper suggested a transplanting model or a 'long bottleneck' to account for the limited genetic variation, rather than a catastrophic environmental change. This would be consistent with suggestions that in sub-Saharan Africa numbers could have dropped at times as low as 2,000, for perhaps as long as 100,000 years, before numbers began to expand again in the Late Stone Age.

TL:DR; According to our observations with animals facing extinction single pair is not enough to repopulate earth. The probability of success is very very low.

SOURCES:
RARE GENETIC UNIFORMITY FOUND IN CHEETAHS
The Cheetah and its race for survival
Inbreeding
Inbreeding depression
What are the general effects of inbreeding

EDIT: These are answers to discussion in comment section.

1)How would this answer given shed light on the non-biblical theory that we are all descendent from one modern human couple of Sub-Saharan African couple?
Ans:The sub Saharan couple mentioned by science are mitochondrial Eve and Y chromosome Adam. They were not a couple, they are supposed to have lived tens of thousands of years apart. They are the most recent common ancestor(MRCA) from whom all currently living people are descended patrilineally and matrilineally.

2) As Mickey Cashen said when a re population effort was made of various mammals (deer, squirrels, etc.) the minimum number of acceptable non-close relatives was 16(couples). Theory says with 2 people the population would die out in 70 years, 4 people 140 years so on.
One related tidbit is that I have read somewhere north america may have been populated by 70 people, I don't know about validity nor I have the source.

3)Question about Humans would have to inbreed  once they evolved to existence.
Modern humans, they didn't come to existence one day,  small small adaptations occurred over time but they were still fertile with previous people. For example one adaptation is loosing of hair.  One individual would have been born with this mutation, but he was still fertile and produced children with similar mutation, gradually the mutation spread and all became hairless. They did not inbreed.
One present time example of adaptation is Lactase persistence in adults, some 35% of global population has evolved into little more modern humans, but they are not different in any other aspect and still fertile with others.

Rinin Rajan
Rinin Rajan, Jack of some trades
Highly unlikely.
Scientists have coined a term called Minimum Viable Population (MVP), which is the minimum number of individual population required for a species to survive in the world. This figure is calculated using computer simulation taking into account inbreeding instability, climatic alterations, predators etc.
And the MVP for humans is estimated to be around 160. But this number is estimated considering controlled conditions, lesser environment fluctuation, basically if you're living like the rich folks in Elysium (Hell of a movie BTW).

160 may work out if your living conditions are like these
Probably not if its like this

Source: “Magic number” for space pioneers calculated

But in reality, this number is calculated for living in the wild, without heavy human interference (both good and bad). So this number would probably be much higher. In other words, risking the entire human population on just 2 people, I would say it's risky bet.

P.S - Watch Children of Men (2006) movie by Alfonso Cuoaron

Edit: Thanks Jonathan for pointing out the error
Adrian Hsiah
I have a feeling it'd be possible but highly unlikely. A lot of answers talk about whether genetically they can do it but I believe the issue is far more practical long before inbreeding and genetic issues would be a problem.

First, you need a suitable couple. The chance the couple is well suited to the task is woefully small to start with.

Firstly, both people are in the same place geographically. Given that all of human technology would very quickly deteriorate, it's unlikely two people would be able to find each other if they're not already in close proximity.

Secondly, both people would almost necessarily have to be highly skilled, knowledgeable and resourceful - well beyond the level of an average human being. Consider just the considerable challenges of a human being surviving in a world where there's no longer public utilities, medical care or modern conveniences (like supermarkets and retail stores). You'd have to know a fair amount of knowledge on how to survive and the skills to use the knowledge.

You'd have to be able to defend yourself against wild animals. You'd have to learn to produce the food and water enough to sustain you and your growing family. You'd have to be able to know how to treat common diseases, heal injuries and deliver babies. Most of us underestimate drastically how many other people we rely on to do the things we can't on a daily basis.

Thirdly, both people would have to be in relatively decent physical and mental shape. It would be all for naught if one of the two humans left on Earth was an elderly person, for example. Or if one was a young child. Or if one was a patient with a terminal disease or with a high risk of a significant condition. Or if either one person suffered from fertility problems. Or heck, if either of them suffered psychological shock from seeing the other 99.999999+% of their species die.

They also have to be of mixed gender, obviously, unless they're sitting on a sperm bank and they're trained in artificial insemination.

Given these issues, the chance that you have two suitable people for repopulation, outside of purposely selecting them for the job, is very slim. Two random people are likely to fail to meet all the necessary criteria.

Next, they have to survive over time while exercise extreme risk aversion. This more than anything ensures that two people repopulating the Earth is improbable simply because the margin for error is so small.

If one is out collecting firewood and a bear comes by and mauls him, the human race is kaput.

If one cuts their hand cleaning a fish and dies from an infection, the human race is kaput.

If the female experiences any complications during any of the childbirths and dies, the human race is kaput.

If either of them contracts any disease that exceeds their capabilities to cure, the human race is kaput.

If either person is crushed by a collapsing building in an earthquake or freeze to death in a snowstorm or whatever... well, you get the idea.

The couple would certainly have to live with safeguards against anything that puts them in any kind of risk.

So a big part of this is that the couple would have to have a suitable base of operations for repopulating our species. secure a place to live where they can grow their family in security, safety and in relative comfort and minimize all risks.

The problem is that where are you going to find a place to minimize the risks while still providing for all their needs? Again, probability wise, they'd need to already be situated in a place with a moderate climate, plentiful resources (including natural food and water sources) and minimal natural dangers. It'd be best if they had access to a self-sustaining, well supplied and isolated bomb shelter for at least a few generations but again, outside of designing this situation intentionally, the likelihood of this happening is slim.

It's also one thing to continuously pump out as many babies as possible and another matter entirely to solve the problems of housing, feeding and taking care of them. As the family grows over the years, you have to consider that you have 2 parents, of whom is perpetually pregnant, taking care of a lot of dependents while trying to provide their basic needs. It's inevitable at some point the family would have to take more risk to grow their family further and supplies they can use will inevitably start to dwindle.

Even if we made it past the first generation successfully, each subsequent generation would almost surely have a higher mortality rate as modern day supplies ran out, machinery started to break down and the need to expand their base increased. 

Ultimately, it's still not impossible but my belief is that it would certainly have to be planned. Two perfectly selected candidates, placed in an ideal location, with a highly automated, low maintenance facility that's self sustaining with renewable power, water processing, farming and the ability to manufacture more supplies. This would enable the human race to at least reach a certain level of stability in population before trying to repopulate the 'natural' way.
Jade McGough
Jade McGough, Software Engineer, B.S. Neuroscience

The "no" answers I see are making several incorrect assumptions: (1) that after a number of generations, inbreeding would surface some deleterious mutation and kill off all future offspring, (2) that two random humans wouldn't have sufficient genetic variance between them, and (3) that endangered humans would react the same way as other endangered mammals. Additionally, the question asks whether repopulation is possible, not whether it's likely or not.

To address these:
(1) During the first generation (the initial two humans), even a serious and lethal mutation (such as Cystic fibrosis) would be passed to 50% of offspring, and those who had it would still have a good chance of living years into adulthood (and being able to reproduce). A much larger risk than a lack of genetic diversity would be the death of the mother from childbirth.
(2) This leads to the issue of genetic variance. The truth is that some degree of inbreeding is relatively common, and rarely very harmful - Assortative mating displayed in humans means that we seek out mates genetically similar to ourselves. Yes, they would probably experience Inbreeding depression over time, but this is something that many isolated, rural populations have experienced, and haven't been doomed by it. Even if it caused lower chances of survival, this wouldn't mean that some generation would be the cut-off for survivability. Humans would lose a huge amount of genetic diversity, but diversity would slowly begin to improve over time from random mutation, which would help to weed out deleterious genes via natural selection.
(3) Endangered animals don't understand that they're endangered, so using them as case studies is not a good comparison. They obviously wouldn't change their behavior in any way to compensate for their endangered status. Humans who were aware of the fact that they were the last ones left would be intelligent enough to reproduce as much as possible.

In our theoretical world of best possible circumstances, a particularly virulent pandemic occurs on earth, wiping out all humans on the planet. Two astronauts (one male, one female) return to earth after several months in space, only to discover that no other humans are alive.

Fortunately, they can scavenge canned food for years and have readily available housing. Thanks to having a library in their town with printed books, they are able to teach themselves basic medical care and farming, and eventually become self-sustainable. Additionally, since human-specific viruses can only survive hours without a living host, most viral disease threats have been suddenly eliminated. The two humans realize that they need to continue the human race, so they decide to have as many children as possible to maintain the diversity that they both offer. They have nine children before infertility sets in. The early generations face many challenges, including comparatively high early mortality rates, but are able to thrive in spite of set-backs thanks to their adaptive intelligence. It takes tens of thousands of years, but over time the gene pool diversifies, and humans spread to populate new towns, eventually migrating to other continents as well.

Therefore, the answer is "yes" - it's possible that two humans could survive and repopulate the earth.

Bruce A McIntyre
Bruce A McIntyre, I love science, and my mind is full of useless information.
/edit2
I continue to be castigated for not "answering the question."  I will now state with specificity... The odds of two young adults, one male and one female, to repopulate the Earth, considering the reality of living on this planet without the convenience of modern science and environment, is less than nil. Nada. Non-existent. It would not happen. We have seen the removal of 99.9% of ALL species that have EVER existed from our gene pool. If our species, (homo-sapiens) were to achieve such a low level, you can be sure that our species would cease to exist. And it would not depend on something so esoteric as genetic drift. It would result from pain, disease, and other predators. We, as a species, are the recipients of incredible luck and good equipment. But we are not unique; others equally suited to their environment have long since disappeared.
edit2/
/edit
Since I have been accused of "not answering the question," let me be specific. Two people (preferably of opposite sexual equipment) could not (in all probability) repopulate the Earth. My answer below is intended to show that the fact that our species continues to exist is primarily one of luck.
edit/

There was a point in time, some 70,000 years ago, when the entire population of our species, homo-sapiens, was down to less than 1,000 individuals, perhaps as few as 100. The extinction of our species was a very likely thing. The fact that the situation changed allowing our species to expand is almost a miracle.
Almost any time a species in the wild gets pared down to a limited number of individuals that species goes extinct.

Here is an article that describes what happened.

Because once in our history, the world-wide population of human beings skidded so sharply we were down to roughly a thousand reproductive adults. One study says we hit as low as 40.

Forty? Come on, that can't be right. Well, the technical term is 40 "breeding pairs" (children not included). More likely there was a drastic dip and then 5,000 to 10,000 bedraggled Homo sapiens struggled together in pitiful little clumps hunting and gathering for thousands of years until, in the late Stone Age, we humans began to recover. But for a time there, says science writer Sam Kean, "We damn near went extinct."
I'd never heard of this almost-blinking-out. That's because I'd never heard of Toba, the "supervolcano." It's not a myth. While details may vary, Toba happened.

Toba, The Supervolcano
Once upon a time, says Sam, around 70,000 B.C., a volcano called Toba, on Sumatra, in Indonesia went off, blowing roughly 650 miles of vaporized rock into the air. It is the largest volcanic eruption we know of, dwarfing everything else...

Robert Krulwich/NPR
That eruption dropped roughly six centimeters of ash — the layer can still be seen on land — over all of South Asia, the Indian Ocean, the Arabian and South China Sea. According to the Volcanic Explosivity Index, the Toba eruption scored an "8", which translates to "mega-colossal" — that's two orders of magnitude greater than the largest volcanic eruption in historic times at Mount Tambora in Indonesia, which caused the 1816 "Year Without a Summer" in the northern hemisphere.

With so much ash, dust and vapor in the air, Sam Kean says it's a safe guess that Toba "dimmed the sun for six years, disrupted seasonal rains, choked off streams and scattered whole cubic miles of hot ash (imagine wading through a giant ashtray) across acres and acres of plants." Berries, fruits, trees, African game became scarce; early humans, living in East Africa just across the Indian Ocean from Mount Toba, probably starved, or at least, he says, "It's not hard to imagine the population plummeting."
Then — and this is more a conjecture based on arguable evidence — an already cool Earth got colder. The world was having an ice age 70,000 years ago, and all that dust hanging in the atmosphere may have bounced warming sunshine back into space. Sam Kean writes "There's in fact evidence that the average temperature dropped 20-plus degrees in some spots," after which the great grassy plains of Africa may have shrunk way back, keeping the small bands of humans small and hungry for hundreds, if not thousands of more years.
So we almost vanished.

How Human Beings Almost Vanished From Earth In 70,000 B.C.

/edit
It is important to remember that extinction-level events such as this are more common than you would think. "More than 90 percent of all organisms that have ever lived on Earth are extinct. As new species evolve to fit ever changing ecological niches, older species fade away. But the rate of extinction is far from constant. At least a handful of times in the last 500 million years, 50 to more than 90 percent of all species on Earth have disappeared in a geological blink of the eye."

Mass Extinction, Mass Die-Off Information, Prehistoric Facts -- National Geographic
edit/
Drew Southern
Drew Southern, I have parents and children

YES.

Here’s the trick: Sperm bank.

The two people are both female, not related, let’s (for good taste) say on the 16th birthdays, and have some basic training in AI.

The Genetic variation would come from the plethora of different donors. I would hope there are gender selection methods for the male gametes (to produce mostly female 2nd and 3rd gen - to avoid inbreeding) though Charles Darwin did perform analysis of incestuous cousin -based relationships and found if it was only occasional then the offspring did not suffer detrimental effects at the level of having 1 set of greatgrandparents in common (6 great grandparents instead of 8).

The scenario gets a bit unsavoury if you mean the two people were the only genetic progenitors. You’d have a “meeting” of Brother and Sister which most higher mammals try to avoid.

I wonder how it would be if it was 2 people BUT the woman was pregnant (shock horror!) with a child fathered by a differrent man? or maybe Heteropaternal Superfecundation Twins?

Mike Louis Griebel
Mike Louis Griebel, Speciation is messy

Everything Nandu Aditya says, but let's take the constructive view. How, with luck, could it happen? By natural means, of course, no genetic hocus-pocus involved. Needless to say this experiment would be deemed unethical. (I have since used the same approach in Mike Louis Griebel's answer to Can there be anything called 'reverse evolution', like humans evolving into apes?)

1. The founders (the man and the woman) would need to be as diverse as possible. Not only that, but their parents should be as diverse as possible. I think the best chance of obtaining that would be to pick four people who spring from an ancient stock and are as pure-bread as possible. As I understand it, some Peruvian Indians contain about 90% of the original Clovis DNA. Off of the top of my head I would suggest we pick one Maori, one Eskimo, one Peruvian Indian, and one from cental or southern Africa. [Joseph Fullerton wrote: "Maori aren't that pure anymore. Just sayin'." — The word is that Africans are much more diverse than all non-Africans put together, so the best thing would probably to pick four very diverse Africans. Fullerton again: "They don't like to be called "Eskimo", it means "He who eats meat raw" in Inuit." — I actually thought about the term "inuit" today in another context. If I had been home 7 hours earlier, I might have corrected it before your comment.]

2. They should have lots of children in every generation. The idea is to milk the founders of their DNA so that as much of their (limited) diversity is represented in the offspring. Everybody probably has one recessive gene that is either lethal or causes infertility; if you inherit the same copy from both your parents, then you are effectively dead for the purpose of the experiment. We want to avoid that. [Dale Prather comments: "Any productive member (fertile or not) of a society/family will increase the odds of that family's survival," which is probably correct. I was focusing on the genetics.]

3. They should ignore the incest taboo for many generations. Relationships between parent and child would be fine. Between children, with your grandchild, with your great grandchild, your third cousin and anything in between is OK for the purpose of this experiment. Random mating might not be systematic enough.

4. Don't cry over lost babies, and do get another as quickly as possible.

A little further down the line:

5. Serial monogamy. Once a woman has become pregnant with her husband, she divorces him and finds another with whom she will have the next child.

Supposing they survive these first generations, they would gradually become more diverse due to random mutations. Eventually they would also discover that incest is not a good thing in general. It would take many generations before their stock could be said to be self-sustaining.

Still, I doubt it would be possible. At any point foreign DNA would help alleviate problems that accumulate.

[Arne Babenhauserheide has some additional suggestions about segregation in a comment.]

Robert Walker
Robert Walker, Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.

Just upvoted Robin Craig's answer to Can two people repopulate Earth? Is it possible?  Makes it clear that it is possible. Thought I'd fill it out a little more with some images and hopefully draw attention to it as it seems there are many answers here being upvoted by people who are not aware of this study and say it is impossible:

The second paper he gives is particularly clear. A pair of mouflon (wild sheep)

One Mouflon Ram

Ovis orientalis - by Jörg Hempel

Mouflon Ram

Mouflon ewe  Ovis musimon by Doronenko

Not those actual individuals - they are for illustration purposes. But just two wild sheep introduced to an island formed an entire population which was not only healthy but increased in genetic diversity. Unexpected heterozygosity in an island mouflon population founded by a single pair of individuals

Humans have the advantage as others have said that they would know that they are an endangered species and would decide what to do about it.

Obviously would have to have a lot of inbreeding. It's possible some humans might decide this is not acceptable and they'd prefer to be extinct. But biologically it seems feasible.

SCENARIO?

The question doesn't give a scenario. Hard to think of something realistic that would kill all humans except for one couple. Or that would lead to everyone dying a natural death except one young couple.

Even an illness that sweeps the world would leave some survivors, including for instance uncontacted indigenous people. Things like artificial life escaping from the laboratories and reproducing in the wild and replacing Earth DNA with XNA based life throughout the world would kill everyone and would still be there at the start of the plot.

I see it as science fiction rather than future possibility. So then in those science fiction scenarios where only two people are left alive, then sometimes they make it so that everyone dies except them in some plague that affects the entire world. They are, say, doing an experiment in living underground for years without any contact with anyone else, including no communication - then when the experiment is over, they come to the surface and find they are the only people left in an abandoned world. That sort of thing. So they have the technology, have books to recover their knowledge etc, but only the two of them.

Or. you can suppose that they are genuis scientists that find the solution to reversing the thing that kills absolutely everyone in the world, but too late to save everyone except themselves.

Another possibility from science fiction is the idea of an aged population where they have artificial ways of giving birth, and most people typically live to some great age like a million, but new children are born, say just two of them. The rest of their society dies of old age. These are the only ones left. And they make a fresh start.

It's one of the most overused tropes in pulp fiction. Adam and Eve Plot - TV Tropes .

Sometimes they are in a world with no technology and many dangers, in which case they probably have little chance, but sometimes they are surrounded by high technology, automated hospitals, libraries. Or maybe are in a starship with all the technology of the starship at their disposal. I'm assuming a scenario like that.

And - a bit more clarification got from a discussion here: How many people are required to maintain genetic diversity?

IDEA OF A MINIMUM POPULATION AND PROBABLE NEED FOR PLANNED BREEDING

The idea of a minimum population is based on the idea of unplanned breeding, where anyone mates randomly with anyone else of the opposite sex, and is a probabilistic thing.

If you can plan who mates with who, then - perhaps one couple has enough genetic diversity, if unrelated originally and they don't have some serious defect for their children to inherit.

They would be able to build up to a population of a few hundred, and then would be able to survive after that, except, that the entire population would be vulnerable to some virus or other condition they are not adapted to until they have survived long enough for some adaptations to arise through actual genetic mutation rather than just shuffling existing genes.

And the example of the sheep would seem to show practically that if you are lucky, then two would be enough if you are lucky with the individuals.

For another example of a species that has survived a major bottleneck, nearly all golden hamsters

Peach the pet hamster

originate from a single litter. Golden hamster

I know it depends on the species as well. You can't draw conclusions from humans directly from sheep or golden hamsters. But these are a few points that seem to have been left out of the top rated answers here, maybe someone with expertise can say more about it?