If you run the risk of something, it means that you are willing to possibly let something bad happen, usually because you don’t have any other option. For example:
- The doctors run the risk of possible side effects of a treatment that hasn’t been clinically tested because all conventional treatment methods have failed so far.
- Politicians carrying out unpopular reforms run the risk of not being reelected, even if the reforms are necessary.
The verb “to risk something” can be used in the same meaning as “to run the risk of something”, for example:
- They run the risk of death.
means the same as
- They risk death.
but “to risk something” is usually used in the opposite meaning, i.e. “to run the risk of losing something”, for example:
- She was risking her health by eating so little. [Eating so little could have had adverse effects on her health.]
The difference between the two is that “to run the risk” means a conscious decision, whereas “to risk” usually doesn’t. You could say, for example, “she ran the risk of hurting herself by eating so little because it was the only way to slow down cancer progression” or “she was risking her health by eating so little, but she didn’t realize it”, but not the other way round.