What is the problem of induction?

Bertrand Russell writes in The Problems of Philosophy, Chapter 6, On Induction:

"We are all convinced that the sun will rise to-morrow. Why? Is this belief a mere blind outcome of past experience, or can it be justified as a reasonable belief?...

It is obvious that if we are asked why we believe it the sun will rise to-morrow, we shall naturally answer, 'Because it always has risen every day'. We have a firm belief that it will rise in the future, because it has risen in the past. If we are challenged as to why we believe that it will continue to rise as heretofore, we may appeal to the laws of motion: the earth, we shall say, is a freely rotating body, and such bodies do not cease to rotate unless something interferes from outside, and there is nothing outside to interfere with thee earth between now and to-morrow. Of course it might be doubted whether we are quite certain that there is nothing outside to interfere, but this is not the interesting doubt. The interesting doubt is as to whether the laws of motion will remain in operation until to-morrow. If this doubt is raised, we find ourselves in the same position as when the doubt about the sunrise was first raised.

The only reason for believing that the laws of motion remain in operation is that they have operated hitherto, so far as our knowledge of the past enables us to judge. It is true that we have a greater body of evidence from the past in favour of the laws of motion than we have in favour of the sunrise, because the sunrise is merely a particular case of fulfilment of the laws of motion, and there are countless other particular cases. But the real question is: Do any number of cases of a law being fulfilled in the past afford evidence that it will be fulfilled in the future? If not, it becomes plain that we have no ground whatever for expecting the sun to rise to-morrow, or for expecting the bread we shall eat at our next meal not to poison us, or for any of the other scarcely conscious expectations that control our daily lives."

The problem simply means that if A has followed(/been associated with) B in all the observation we have made uptil now, then it is not necessary that A will follow B in all the observations we make in future as well. Consider the example of a stone falling down. Every common man believes a stone dropped will always fall towards the earth, but is this belief rational? Let us inquire: why do we believe that a stone will always fall towards the earth? The answer is that because we have seen this happen millions of times. True, we have experienced a stone falling to the ground many times but we have never experienced the stone falling to the ground always. [We have seen it falling to the ground always in the past but we haven’t yet experienced it falling every time in the future.] Similarly, why do we believe that the sun will rise tomorrow? It is because we have seen it happen always in the past. But just because an event has always taken place in the past doesn’t mean that it will certainly happen in the future.

It would be interesting here to narrate that in the time of Hume in 18th century, one of the examples used to describe problem of Induction was: Just because all the swans we have observed are white, does it means that all swans are white? And then, as if nature decided to participate in this philosophical puzzle, the black swan (Cygnus atratus) was discovered in Australia... a real life instance of the problem of Induction in action!