Question: Suppose the winning combination consists of $7$ digits, each digit randomly ranging from $0$ to $9$. So the probability of $1111111$, $3141592$ and $8174249$ are the same. But $1111111$ seems (to me) far less likely to be the lucky number than $8174249$. Is my intuition simply wrong or is it correct in some sense?

Answer: 
Another argument in favor of choosing a random-looking number is the following: suppose 1111111 is drawn. The scientifically uneducated audience will likely complain that it "can't be random" and something went wrong (or maybe that you cheated), and in the end they'll have the draw cancelled or repeated.

Sadly, there's no arguing against that --- I bet you can convince the average judge and jury that "1111111 is not random".

(That said, in real life 1111111 *is* indeed more likely to appear than 8174249, since a mechanical or programming error in the drawing machine could make it more likely to have repeated numbers drawn than completely random ones).

In short, real life is not like mathematics. :)
The intuition might be wrong for the following cause:
You compute the probability of the event "**any** special number is extracted" and is X.

Knowing first probability you might be tempted to choose from the "special set", thinking that the probability to win will be higher. What you forget is that you will have 100% probability to lose in all cases that the number is not in special set, combined with some probability to win from the special set.

Combining these probabilities will still give you 0.0000001. The same will happen if you choose from the "not special set".