The difference between may and might is very small:
Use may when the event is slightly more likely to happen:
▶️ “What are you doing this weekend?”
“Shopping! I’m going to buy some new clothes, and I may get a new hat as
well.” (it’s slightly more probable that I will buy the hat)
▶️ ”What are you doing this weekend?”
“I might go to the movies. I’m not sure.”
(it’s slightly less probable that I will go to the movies)
However, in this simple case, the words really are interchangeable; you can use
either one.
When making guesses about something that happened in the past, we usually use
might + have + past participle.
▶️ “Why is Sheila so happy today?”
“I don’t know. She might have gotten a promotion – I’d heard a rumor that
the boss was thinking of making her a manager.”
When asking permission, use may (or can/could):
▶️ “May I open the window?”
This question is correct, but it sounds rather formal. Most English speakers
would probably say “Can I open the window?” or “Could I open the window?”
May not means “no permission”; might not means “maybe not.”
▶️ Students may not use cell phones during the exam.
= using cell phones during the exam is prohibited
▶️ I might not go to the wedding.
= maybe I will go, but maybe I won’t go.