May / might

mayWe can use 'may' to ask for permission. However this is rather formal and not used very often in modern spoken English

• May I borrow your pen?
• May we think about it?
• May I go now?

We use 'may' to suggest something is possible

• It may rain later today.
• I may not have time to do it today.
• Pete may come with us

mightWe use 'might' to suggest a small possibility of something. Often we read that 'might' suggests a smaller possibility that 'may', there is in fact little difference and 'might is more usual than 'may' in spoken English.

• She might be at home by now but it's not sure at all.
• It might rain this afternoon.
• I might not have time to go to the shops for you.
• I might not go.

For the past, we use 'might have'.

• He might have tried to call while I was out.
• I might have dropped it in the street.