The statement that the Sun will probably rise tomorrow can be proven in the same way that we can prove that the last marble in an urn will turn out to be red, because all the other marbles we have already drawn from the urn have been red. The more red marbles we draw from the urn, the more probable it is that the next marble we draw will also be red. The we expect the unobserved to resemble the observed. The question of the justification of induction is whether that expectation is justified, and to what extent it is.
I'm not sure what Russel intended but ...
This example with the urn and marbles has another insight to offer. From the evidence we have so far (we have retrieved only red marbles) it is a "natural" conclusion that subsequent marbles will also be red. However this is where the evil of plausibility (that is the human emotional
experience of truthishness) gets us in trouble.
For an urn filled halfway with white marbles, then filled the rest of the way with red marbles, our intuition gives an answer directly opposed to reality. In this more specific scenario the more red marbles we've seen so far, the less likely it is that we will get a red one on the next pull.
Now this is obviously an abstract scenario, but the same principle applies to all inductive reasoning of the "sun rising tomorrow verity." The point here is that cataloging the past does not necessarily tell you anything useful about the future at all (not even in a statistical sense.) However we tend very strongly to believe that it does, often to our own detriment.
Deductive reasoning is not of much use in cases like this though. As we do not know how the urn was prepared, the only answer deduction can lead us to is that we do not know at all what color the next marble will be. The same is true for the rising of the sun, if all we know about it is that it rised every day of our lives. However, via the long history of deduction based scientific investigation, we can apply newtons laws to a model of the solar system and say with great certainty that the sun will rise tomorrow.
Another problem with induction is that a "satisfying" conclusion that has been arrived at via induction can, and very often does, halt further inquiry which may have lead to some enlightening idea down the road. History in fact provides an excellent example with regard to the sun rising issue (Galileo and Pals.)
I hope this was helpful;
(As an aside: I mentioned our plausibility emotion, because people generally accept inductive arguments when they think the answer is plausible to them. In other words they "like the idea." It's an excellent way to manipulate people if you're pushing an agenda. All sorts of social institutions use it to great effect daily.)
[edit: semi-developed thoughts from later in the evening beyond this point]
Statistics is the mathematical equivalent of induction and is prone to the same basic problems. Perhaps that's why Russel refers to probability? (I assert they are equivalent in a deep way, that is: that induction is our minds' implementation of statistical mathematics; or more accurately a sub/superset.)
Induction, and statistical methods both are
powerful tools for leading us towards interesting realms in which to apply deduction. This for me is their "proper" place in the order of thought, but I seem to be in the minority on that assessment. A great many people seem to value their intuition above any form of deduction ("confirmation bias" etc. if someone is interested, you'd find it in the psych section of your local internet.)
These methods are also better than nothing. If you don't have the information, or tools to get the information, you'd need to apply deduction rigorously, then intuitive methods may get you someplace other than nowhere. These sorts of problems are very common, in fact these categories of problem are the most common of those we encounter day to day.
Perhaps humans are "optimized" for induction for these reasons. Which would give me a satisfying answer to my recurring pseudo-existential crisis of why people act remarkably dumb most of the time, even those folks who seem otherwise intelligent (I would consider myself an example pretty often.)