First, the chance of this is absolutely minute. Asteroids or comets of diameter 10 km or larger hit Earth only once every 100 million years. We have already found all the near earth asteroids of diameter 10 km upwards and none will hit us this century. So that does indeed leave comets - or the very rare asteroids in long period comet like orbits, as the remaining possibilities. There’s no consensus on what percentage of impacts are by comets but far fewer than one every 100 million years. That means the chance of such an impact in this century, anywhere on Earth, is far less than 1 in a million. It can happen - Mars had a close flyby by comet Sliding Spring, and though Mars gets more impacts than Earth does, Earth must have a chance of being hit by a large comet too, but it is very very unlikely.

Yes, in the exceedingly unlikely case of such an impact, the effects would be devastating as the other answers say. The maximum impact velocity for a long peiod comet orbiting the sun at Earth’s distance from the Sun is 72 km / sec, and you can feed this figure into the Imperial College London Earth Impact Effects Program for an approximate idea see Calculated Results (this shows the effects at the opposite side of Earth from Australia) and for effects on Australia, 1000 km from the impact, see Calculated Results - basically anything easily combustible burns.

So, I think it’s fair to say that few humans or other large terrestrial mammals would survive in Australia due to direct heat and fires over 1000 km from the impact. An impact this large sends ejecta into space with more than escape velocity, so there would be secondary impacts of smaller lumps of rock hitting over the entire Earth, though not much would get as far as the opposite side of Earth. There might be firestorms world wide due to these secondary impacts, with most trees burning. The worst effects would be the long term “nuclear winter” effects leading to much reduced crops for several years.

But though many would die, I don’t think humans would go extinct. First, for the impact event itself, even if there are global firestorms, you just need a few to survive, in submarines, for instance (storms, even tsunami are surface effects and submarines don’t experience them unless close to the surface), or in a base in Antarctica or some cold desert with no trees or other vegetation to be set alight.

And then - the thing is that humans are omnivores that can survive with minimal tools anywhere from the Arctic to the Kalahari desert. We can eat fruit, roots, shellfish, animals, birds, almost anything edible, There would be some stores of food survive the event also. We would surely not lose the ability to make primitive boats, to travel to other parts of the world. We are mammals also and not dependent on sunlight to stay warm. When the dinosaurs went extinct, birds, turtles, small mammals, many creatures survived. Even if 90% of all species went extinct, humans would surely be amongst the 10% of survivors, it seems to me.

That’s even with no preparation, and new comets are usually discovered one to two years before they fly through the inner solar system, so we’d have at least a year of warning too. So we’d prepare, we’d set up shelters, store food, move to parts of the world that would not be affected by a firestorm if that was predicted, or build fireproof shelters, maybe with oxygen also if it’s predicted that the fire would use up the oxygen at ground level for the duration of the fire, etc etc.

The whole thing though is such low probability that it’s not worth spending much time thinking about. Movies often focus on large dramatic impacts, sometimes far too large to be plausible at all (if you can see the impactor next to the Earth with both shown to the same scale, in HD, it is far too large to be plausible, larger than any known impact for the last over 3 billion years in the inner solar system from Mars inwards). But the smaller impacts are far more likely.

By probability the first successful predicted impact of a hazardous asteroid is far more likely to be a few tens of meters in diameter, perhaps up to 100 meters in diameter than a big asteroid kilometers in diameter. Those smaller impacts happen every few thousand years, and the smaller ones every few centuries, or even more often. Also most of Earth is covered in sea or is uninhabited desert. So don’t expect this headline

“Large comet to hit Earth, size of New York, crater two hundred kms in diameter, global effects, ”

It’s possible but extremely unlikely.

This is far more likely:

“Large asteroid to hit sea, 50 meters in diameter, co-ordinates … on date and time … - ships and planes should avoid that area at the time of impact”

or

“Large asteroid to hit remote desert, 50 meters in diameter, co-ordinates … on date and time … -”

and a much smaller chance, but still possible, perhaps would happen every few millennia:

“Large asteroid to hit inhabited area, 50 meters in diameter, will hit at co-ordinates … on date and time - inhabitants need to evacuate … -”

And even that is low probability and may not happen for several centuries, or millennia. The Tunguska event would have been devastating if it hit a city, but it hit a remote uninhabited area, and that’s the most likely outcome. It’s by far the largest impact event to be recorded in human history (Meteor Craterr in Arizona probably predated the Hopi Indians although they have legends about its formation)

Effect of the Tunguska impact. Tunguska event - result of airburst of a small comet or asteroid (evidence points both ways) - devastated 2,000 km2 of forest, diameter of asteroid or comet f 60 to 190 metres (200 to 620 feet), hit an uninhabited area of Russia on the morning of 30 June 1908. Something like this hitting a desert or uninhabited region or the sea is far more likely, and wouldn’t have any global effects. It might well not harm any humans as in the Russian case.

If it does happen and we have enough notice, like a few weeks, we can evacuate the impact zone, If we have more notice than that, years or decades, we can deflect it quite easily, or break it apart.

What we need most here is knowledge. It’s the one natural disaster we can not only predict to the minute, we can also prevent it too. The way to achieve that is to know the orbits of all the small asteroids so well we can predict them decades in advance. We are doing ground based searches to achieve this, but it can be speeded up hugely with space telescopes - the Sentinel telescope project would cost 0.5 billion and many military projects to protect us against each other cost many times that. Any of the more wealthy nations on Earth could finance this using a small fraction of their military defense budget. Many private individuals could also if they felt it was important enough. We can also improve our detection of comets, but right now the priority is to detect the smaller asteroids in shorter period orbits as that’s the highest probability risk, so you can reduce the risk considerably for less cost that way.

With a space telescope orbiting between Earth and the Sun we could map out most even of the smaller asteroids down to 20 meters in diameter in less than a decade, for less than 1/400th of the predicted lifetime cost of the renewed Trident nuclear submarine which our government in the UK recently approved. We could send more than 400 such telescopes into orbit to detect these asteroids for the cost of our nuclear deterrent. And that’s just one country. The world can certainly afford the cost of these space telescopes collectively, and though the risk is tiny, it’s not much to pay for such return. That’s 7 cents per inhabitant, one off payment, for the entire world, or less than $2 per inhabitant for the US, or $1 per inhabitant for the EU, one off payment.

Meanwhile the UK Trident nuclear deterrent renewal lifetime cost for the UK is ab0ut £3,000 per head of our population of 64 million ($4,000) based on that £205 billion figure. Do we have our priorities right? (Replacing Trident will cost at least £205bn, campaigners say - that’s before the pound devalued)

The one in a million chance per century for a 10 km or larger asteroid impact is a minute chance for anyone individually, way below any ordinary risks, far more likely to be struck by lightning, but when you multiply by the population of the Earth it’s several thousand people expected number killed in a century.

However given that we have already found all the 10 km diameter NEOs, it’s more like a 1 in 100 million chance since only 1 in 100 impacts are by comets (rough estimate). Now that we’ve completed the search for 10 km NEOs, the risk of death by impact is now roughly equal between the smaller and the larger asteroids, and the risk from the larger asteroids continues to decrease as we find the 1 km asteroids too. We know how to find those and have found 90% and will find 99% some time in the 2020s but with the smaller asteroids, the existing programs won’t find most of them until much later which is why we need to expand on those programs with space telescopes to meet the target of finding 90% of all hazardous asteroids by the 2020s.

See also my Giant Asteroid Headed Your Way? - How We Can Detect And Deflect Them