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: 
You should never bet on that kind of sequence. 
Now, every poster will agree that the odds of any sequence from 000000000 through 999999999 has an equal probability. And if the prize is the same for all winners, it's fine. But, for *shared prizes*, you will find that you just beat 10 million to 1 odds only to split the pot with dozens of people. 
To be clear, the odds are the same, no argument. But people's bets will not be 100% random. They will bet your number as well as a pattern of 2's or other single digits. They will bet 1234567. I can't comment whether pi's digits are a common pattern, but the bottom line is to avoid obvious patterns for shared prizes.

When numbers run 1-50 or so, the chance of shared prizes increases when all numbers are below 31, as many people bet dates and stick to 1-31. Not every bettor does this of course, but enough so shared prizes show a skew due to this effect. 

Again - odds are the same, but human nature skews the chance of split payout. I hope this answer is clear.
A couple of people have commented on how to increase the odds that you won't have to *share* your lottery winnings with other people, so it's worth mentioning a book on precisely this: [*How to Win More*](http://rads.stackoverflow.com/amzn/click/1568810784), by Norbert Henze and Hans Riedwyl. Here's a brief review, written by David Aldous:

> 
> Despite the title, this is a well written and serious book on the
>  modern "pick 6 numbers out of 49" type of lottery. Of course you
>  can't affect your chance of winning but you can try to choose
>  unpopular number combinations to maximize your share if you do win.
>  Uses empirical data from around the world to describe "foolish ways
>  to play" (based on previous winning or non-winning numbers, patterns,
>  etc) -- what makes these foolish is simply that too many other people
>  use them. Concludes with a non-obvious recommendation: choose randomly
>  subject to several constraints (one of which requires a bit of math to
>  understand: a quantified measure of non-arithmetical-progression). An
>  appendix has some upper-division college level math probability
>  analysis, but non-math folks can just ignore it.
> 
> 
>