Our 3 year old has done a few things that were risky, e.g. she stood up on a chair with wheels to reach something. I explained that it is very dangerous and that she could fall. Her response - as has been with a few cases like this now - was "No, I don't fall." and my efforts to try to explain that she could fall don't seem to be working. Am I expecting too much at this age when I phrase things in the context of risk/chance/odds?
On an instinctual level very young children understand risk. What I've noticed in my son (under 2): fear of being burnt, fear of falling from a height, fear of being underwater, fear of scurrying insects, fear of unknown animals/people, and some small fear of the dark.
I think you are correct in thinking that young children don't know risk/chance/odds. Some adults know little of it as well. Maybe a reminder that certain behaviors might might result in getting hurt. Some phrases we use:
These are trying to associate behavior with pain and to teach language. Ouch and hot are so concrete that they communicate a lot. Fall, hurt, nice, and careful are harder.
There is also a thrill and joyfulness associated with being on the edge of fear. A sence of accomplishment when we overcome our fears. Finding our limits may require some amount of injury. We want our children to avoid all of the cuts, scrapes, and breaks possible but how are they going to learn to take risks if we always intervene. The article below talks a bit on this.
Can a playground be too safe? via New York Times
There are a range of dangers that seem to require a child to experience before they consciously understand risk. For example - fire. Children usually love to look at fire, and it warms them so they don't inherently fear it, which is a big worry for parents.
Our solution was to tell them it was sore, and then to let them reach towards the fire (while loosely holding their hand, long enough that they could feel it was sore, and then pull their hand back quickly, while stressing "OUCH!"
Each of them only needed that once - and never tried to go too near a fire again.