'Young patients may learn which diseases they could get when they are older.'
In the above sentence, How much possibility does 'may' mean?
According to Webster's 4th ed:
"Possibility - the quality or condition of being possible"
"Possible - that can be; capable of existing"
This is an on or off word. Something is either possible or not. "May" connotes that it is possible, but conveys nothing of "will" or "likely-hood" or "probability" - which was probably the word you are looking for in "How much possibility..."
"May" in this context conveys that it is possible or that they will have the option if they so choose, or should circumstances align in a way that they do.
Replacing "may" with "shall" or "will" implies 100% probability that this will occur. To say "will likely" instead actually implies a probability somewhere greater than 50%, or to say "will not likely" somewhere less than 50%.
In short: "may" does not imply any probability, but declares it is possible. Additionally, possibility has no metrics other than on or off, yes or no.