In several contexts (not always), /possibility/ has a positive meaning, like:
"answer to that insertion, seems a possibility to you";
You may stick a number to /probability/, usually like a percentage
(in this case 100 % means: sure event), or as a fraction of unity (in this case 0.5
means an equal chance of the event to verify or not).
You can't stick a number to /possibility/.
The probability of rain within the next 5 hours is 95 %
The probability of rain tomorrow morning is 65 % from the data available now.