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.
As my Math teacher used to say, "I don't know the way to win more often in lotery, but I know the way to win MORE":

In fact, in a situation where the gain is shared between the winners, you have your interest in taking numbers that, if you win, should have less winners possible. Actually, people often choose their birth date, so if you avoid combinations like 19xx and 01-31 when you choose your numbers, you're more likely to not to share your gain with somebody wif you win :D