This was a conversation I was having with a colleague earlier today. He was offered a postdoctoral position and doesn't want to move after that. It's a very highly regarded university so it's understandable that he doesn't want to leave, but it made me wonder how common this is. Do postdocs frequently advance in the ranks at the same university or is it still the norm to go to school A for PhD, school B for postdoc, and school C for professorship?
It's not common to stay at the same university after doing a postdoc. When you do a postdoc somewhere, you can certainly apply for a faculty position there afterwards, the same as anyone else, and you might get it. The fact that the department is familiar with you and your work could be a modest advantage, but there may be disadvantages as well. For example, the university already has someone in your general area, namely your postdoctoral supervisor, so they may feel less need to hire you as well.
Without some special reason to think otherwise, the chances of being hired at this university are probably roughly the same as they would have been had you done a postdoc somewhere else. That generally means they're low. In most cases, you can't just pick a university to focus your attention on and hope to get a faculty job there. Even if you're qualified, many other applicants will be as well, and some may be even more so. Of course you may be exceptionally accomplished, or in a hot field in which recruiting is very difficult. If that's the case, then you may be in a strong position to stay, but otherwise it will take a lot of luck.
Another way to look at it is by counting. Most major research universities hire a lot more postdocs than faculty members. (An average faculty member will supervise quite a few postdocs over the years, but the university isn't growing exponentially.) Probably most of the postdocs would like to stay, but there just aren't enough faculty positions to hire them, even if nobody were hired from outside. In other words, it's the same argument as for why most graduate students can't expect to find faculty positions at institutions comparable to the one they graduated from.