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: 
Your intuition is wrong. Compare the two statements

A. The event "the lucky number has all its digits repeated" is much less probable than the event "the lucky number has a few repeated digits"

B. The number 1111111 (which has all its repeated digits) is much less probable than the number 8174249 (which has a few repeated digits).

A is true, B is false.

BTW, this can be related to the "[entropy](https://physics.stackexchange.com/questions/66651/clear-up-confusion-about-the-meaning-of-entropy)" concept, and the distinction of microstates-vs-macrostates.
As mentioned by all other people here there is no disadvantage in picking 11111111 as your lottery number.

However I would just like to add that there is a pattern in these kinds of thinking mistakes.

Transitivity
============

Let's say that :

* A = a very unlikely event
* B = a very unlikely event

**Our brain is tempted to use transitivity whenever it can.**
Immediatly we want to know what happens if A and B would happen at the same time. Usually, the chance of A and B happening at the same time would then be even smaller. We would have to multiply the chances.

Cards, An example of transitivity
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When taking a random card from a deck of cards (52 cards):

* The chance to take an Ace = 1/13
* The chance to take a Spade = 1/4
* The chance to take the ace of spades = 1/13 \* 1/4 = 1/52

Here it makes perfect sense.

The lottery problem , transitivity ?
====================================

It is tempting to apply this reasoning to the lottery problem.

* A = the chance to guess the lottery number is very small
* B = a number with all equal digits seems like something rare

And here again, our brain is tempted to use transitivity
The reasoning would then be that ...

* the combined chance of (A and B) is smaller than the chance of A.

Unfortunately this is incorrect. You cannot apply transitivity here.
But why not ? I will illustrate with some more examples.

Dice problem , another transitivity failure
===========================================

Here is an obvious example that makes the same mistake:

* the chance to throw a 1 with a dice = 1/6
* the chance to throw a 2 with a dice = 1/6
* What is the chance to throw a 1 and a 2 at the same time with the same dice ? Is it 1/36 ?

It is of course impossible to throw a 1 and a 2 at the same time.

When can you apply transitivity ?
=================================

So, what is the difference between the dice and the cards?
The difference is that (as with the cards) **when applying transitivity we have to make sure that the individual conditions that we are combining do not interfere with each other.** If there are no such dependencies then you can blindly apply transitivity and multiply the individual chances.

Applying it !
=============

There is 50% chance that I am a man, there is 50% chance that I am a woman. However, the chance that I am both is not just 25%. This is a perfect example of interfering conditions.

So, yes a digit with all equal numbers is something special (0.00001%), and yes guessing the lottery number is difficult (0.000001%). The problem is that you are trying to combine these 2 conditions, but the conditions are interfering. So in conclusion, even while it seems that you should multiply the chances (0.0000001 \* 0.00000001), this is in fact a mistake.

I hope this explanation gives you a better insight.