What is the likelihood of getting tenure? What fraction of new assistant professors end up receiving tenure at the institution that they started at? This of course varies by field and rank, so please include the area and range of ranks you are thinking of when you answer. (i.e. "Computer Science, top 25 but not top 10", etc.)

It seems hard to find hard statistics on this question. When you get official statistics for how many people come up for tenure and are denied, you get an overly rosy picture, because people who are not going to get tenure often leave before they come up for it.
This question was the topic of a recent article in Science. Briefly, around 50% of tenure-track faculty will actually remain to become a full professor. This varies with gender, discipline, and year (e.g., its harder to obtain tenure now than it was 20 years ago). I strongly recommend reading this Ars Technica article, which explores the aforementioned Science paper in depth.