Odds of Getting In
Out of a population of 300 million people, there are only 1,500 or so licensed jockeys in the United States—so yeah, not a ton of people get into this career (source).
Then again, it takes a very specific fit, in terms of body type and the desire (or death wish) to actually do the job, so competition isn't exactly fierce. If you get into jockey school, the question is less about whether you can get this job and more about whether you're willing and/or nutso enough to actually want it.