possibility vs. risk

Possibility denotes an assertion that some specific beneficial or wanted condition has some chance (made specific or not) of occurring. That chance may (or may not) depend upon something being done first in order to create that chance. 

Risk (in contrast) denotes an assertion that some specific costly or unwanted condition has some chance (made specific or not) of occurring. That chance may (or may not) depend upon something being done first in order to create that chance. 

Examples:

  • "If you take this medicine, you have a 23% possibility of your condition improving, a 70% chance that it would continue to deteriorate as if you didn't take the medicine, and a 7% risk that things would get worse even faster." (In this example, you could call the "70% chance" a "70% risk" if you were contrasting it with the 23% possibility of improvement. Or you might call it "70% possibility" if you were contrasting it with the 7% risk of your condition deteriorating even faster than without the medicine.)

  • "There's a 17% risk of high winds today." (This risk exists regardless of what anyone does). Although some people might consider it to be "a 17% possibility" because they would value high winds for some reason.

  • "If he chose courage to ask her on a date, there is a possibility that she will go on a date with him. If he doesn't ask her, then the risk that she will not go on a date with him is 100%." Many of us would not think of calling the chance of her not doing on a date with this man if he doesn't ask her a risk. But since there are the two alternatives of asking or not asking, then it's appropriate to use the word risk (instead of chance) since, considering these alternative conditions, "going on the date" is considered beneficial to the man and "not going on the date" is considered costly. Although it's also valid to say it is not costly because his "normal" condition has not gotten worse.