are you in the US or UK? I can only speak for the US.
You question of the probability of a career ending injury or disability is a good one. And when trying to answer it, you'll find the inherent problem of finding a good policy. I've always referred to it as "disability insurance" But, the fundamental question is, what does it mean to be disabled, or as you put it, a career ending illness? Opposed to life insurance, where it is rather cut and dry, disability insurance is a huge gray area. For example, say you're a surgeon and in your upper 30's and your hand becomes unsteady. The hospital picks up on this and won't let you operate. You're healthy and in great condition, but you can no longer do what you spent all these building up to do. So, are you disabled? We can all go back and forth all day about yes or no, but what really matters is what the policy says. So in essence, not all policies are created equal.
First, however, you need to decide what your plan is if you can no longer work because of sickness or injury. Do you move back in with your parents and mooch off of them the rest of your life? Perhaps you have a partner/spouse who will work and take care of you? Will a gubimint program take care of you? But, the important thing is to not buy the insurance unless you need it. Certainly, if you die, you don't need to worry about shelter and how to feed yourself. But, if you become disabled, you're still here and you need to worry about how to keep yourself here. Being "single" you probably only need a little, but that is for you to decide. Obviously, if you had dependents, it becomes a bit more clear. While it is true it will be a lot cheaper if you buy when you're younger, you'll save money only buying insurance in the appropriate amount and when you need it.
Then you need to decide what policy covers you the best. In the US, Guardian is considered to be the "gold standard" and who carries my disability policy and most policies for people like our surgeon above. I believe mass mutual also has a pretty good policy. They'll ask you if you have any disability policies at your employer, and the answer is most likely you have a crappy policy. You cannot be insured over your current compensation.
Anyway, if you see there is a need, there is a ton more information out there. I believe I have a post on my neglected blog which details the neat things guardian throws out there in their policy. I can answer questions as well, but ultimately, you'd probably be better served not buying a policy from your financial advisor or anyone they'd recommend. You'll need to find a good insurance agent and they'll walk you through the process. I know there is a lot of overlap these days, but I prefer a dedicated insurance agent.