Odds of Getting In
If you're one of the 45,000 people each year who graduate from law school, you will likely have a window from age 40 to 60 to potentially be appointed to the Supreme Court. (A candidate under 40 is considered too unseasoned for the position, and a candidate over 60 is considered too close to dying and allowing the next President to put his guy in as a replacement.) During this 20 year stretch, there will probably be four or five vacancies on the Court. If we throw out a few thousand people to account for career changes, disbarment, and death, you're looking at roughly a four in 800,000 chance, which we can reduce to one in 200,000.
In reality, though, this is the sort of gig where you have a pretty good idea what your chances are going in—for 99.9999% of us the chances are literally zero. For those with the appropriate credentials and an actual shot, it becomes an entirely different ballgame based on the makeup of the Court, the preferences of the current President, and various other factors. For example, we've heard that sporting a mohawk at the confirmation hearing is a bad idea.