the present:

Paul may be home  (Perhaps he is at home now or perhaps not) 

  compare the example above with: Paul is at home (The speaker is sure about what he is saying.)

   the future:

Paul may arrive late.  (Perhaps he will arrive late or perhaps not.)

  compare the example above with: Paul will arrive late. (The speaker is sure about what he is saying.)

MAY AND MIGHT + perfect infinitive are used to speak about possibility in:

  the past:

James may have finished his homework.

  According to some grammarians, the use of might instead of may implies that the speaker is more doubtful about the likelihood of something happening than if he used may:

Paul might be home (The speaker thinks Paul is less likely to be home than if he said, Paul may be home.)
 
Paul might pass his final examinations. (The speaker is more doubtful about Paul's chances of passing his exams than if he had used "may.")

 

  Negative form: remember the general rule  (...aux.verb + not + ...): 

He may not have finished his homework.
He might not be at home

  Interrogative form: When "may" and "might" are used to express possibility, they are not usually used in the interrogative form. The following expressions are used instead:

• Do you think...?       Do you think Paul is at home?
  
• (be) likely:     Is it likely that he will stay for two weeks? / Is he likely to stay for two weeks?

 
COULD can be used instead of "may" or "might"  to express possibility.

Jonathan could know the answer to your question. (Perhaps he knows the answer to your question or perhaps not.)