may is used to express permission
May I come in? Yes, you may.

may is used to express something which is possible but not certain
He may be hiding a weapon.

may not is the proper negative form for may

might not are the proper negative form for might

might is the past tense of may
She might have done better if she had studied

might is used to express something which is unlikely
The boss might ride a tricycle around the office, but I doubt it.

might is better to use than may concerning doubt about something happening
We might not attend to the event.

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