Further to that, "may" often (not always) carries connotations of choice and "might" of trying.
So for instance:
"If I choose to, I may become a doctor. Otherwise I may go into law or teaching." (The ability is not in doubt, only the choice)
And
"I might become a doctor, if I can pass all the exams." (The ability is in doubt)
Or to put it another way with a slightly different emphasis:
“OK, you may become a doctor.†- i.e. you are allowed to become a doctor if you want
“OK, you might become a doctor.†- i.e. If you are able to achieve it, you could become a doctor.
(That isn't the only way of reading any of those sentences and, as Tramp said, both can mean a possibility with "may" being more likely than "might").
So for instance:
"If I choose to, I may become a doctor. Otherwise I may go into law or teaching." (The ability is not in doubt, only the choice)
And
"I might become a doctor, if I can pass all the exams." (The ability is in doubt)
Or to put it another way with a slightly different emphasis:
“OK, you may become a doctor.†- i.e. you are allowed to become a doctor if you want
“OK, you might become a doctor.†- i.e. If you are able to achieve it, you could become a doctor.
(That isn't the only way of reading any of those sentences and, as Tramp said, both can mean a possibility with "may" being more likely than "might").