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: 
Most of the answers are making two assumptions about the nature of the lottery being played. Firstly that order matters (that a drawing of "17, 23, 31" is not the same as a drawing of "23, 31, 17"), and secondly that balls are replaced (that you put back the "17" ball after it's been drawn, before you pick the next number).

Depending on the lottery being played, one or both of these assumptions may be wrong. Suppose you have balls numbered 1 through 49:

* If order matters and balls are replaced, then "1, 1, 1" is as likely as "1, 17, 23".
* If order doesn't matter and balls are replaced, then it depends on how many of each number there are. If there's one of each number, "1, 1, 1" is six times less likely than "1, 17, 23", as there' six different ways to make the latter (since "1, 17, 23" and "23, 17, 1" are the same), but only one to make the former. If there's three balls of each number, they're equally likely.
* If order matters and balls are not replaced, then it depends on how many balls of each number there are. If there's one of each number, "1, 1, 1" is impossible. With three of each number, "1, 1, 1" is still less likely than "1, 17, 23", as, e.g. for the second number, there's two "1"s but three "17"s. (More specifically, there's six ways to draw "1, 1, 1" (3\*2\*1), but 27 ways to draw "1, 17, 23" (3\*3\*3)).
* If order doesn't matter and balls are not replaced, then it depends on how many balls of each number there are. If there's one of each number, "1, 1, 1" is impossible. If there's three of each number, "1, 1, 1" is *way* less likely than "1, 17, 23" - there's six ways to draw "1, 1, 1", but 162 ways to draw "1, 17, 23" (9\*6\*3).

For example, in the UK national lottery, there is one of each number (from 0 to 49), balls are chosen without replacement, and order doesn't matter (giving a total number of possiblities of 49Choose6 = about 14 million). So "1, 1, 1, 1, 1" is impossible, and "1, 2, 3, 4, 5, 6" is as likely as "1, 3, 15, 27, 41"
In canada, lottery tickets are manually drawn and each number cannot be drawn more than once.

For instance, Loto 6/49 (6 numbers, 1 to 49), an "human instution" inprobable number would be 1-2-3-4-5-6 (6 number in a row) and I beleive this is less probable then a more general number like 1-11-20-30-31-45.

Why? Because of the way it's drawn. If the number were drawn all at once by a computer for instance, I wouldnt beleive this. However, it's not the case in this example. Each time a number is drawn, it gets removed from the pot and the pot is scrambled again. Thus, there less numbers in the first 10 digits after each draw, making them less likely to be drawned. 

Call me crazy, but I would never bet my money on a ticket that has 6 numbers in a row.