We use may to express possibility, permission, and wish.
- A suicide bomber may have caused the explosion. (Possibility)
- May I have another lollipop? (Permission)
- May you live to the ripe old age of 100. (Wish)
We use might to express possibility, permission, suggestion, and annoyance.
- I might have stepped on the banana skin and fallen if I had not noticed it. (Possibility)
- Might I ask why you two are not speaking to each other? (Permission)
- You might like to have a look at my two new pets – a turtle and a guinea pig. (Suggestion)
- You might at least try to look like you were enjoying the soup. (Annoyance)