may / might
Two modal verbs of possibility and permission, and the difference between may have and might have.
For an open possibility in the present or future, may presents the outcome as more likely, and might presents it as less likely: The ferry may be late is the firmer warning; The ferry might be late is the more tentative one.
Like the other modal verbs, may and might do not inflect: they have no -s form, no infinitive, and no participles. Might is not the past tense of may; it expresses past time only in reported speech or thought, after a past-tense verb: Elena thought the store might close early. Might also belongs in the result clause of an unreal condition: If the rent fell, Elena might stay.
Both may have and might have present a past event as possible while the facts remain unknown: The package may have arrived while the office was closed. Only might have is counterfactual: it describes an outcome that the facts have ruled out: Without the extra pump, the basement might have flooded. May have there would wrongly imply that the basement possibly did flood.
Where may could be read as permission, especially with not, use might: The interns may not attend the briefing can forbid them; The interns might not attend the briefing only predicts.
Use may both to grant permission (You may go.) and to ask for it (May I borrow your stapler?). Can for permission is standard English and is not an error, but many people still criticize it as casual or imprecise and judge the speaker who uses it. Wherever that judgment matters, as in a job interview or a request to a client, use may (see can / may). Might I asks with more deference than May I (Might I borrow your stapler?), but in American English it is rare and sounds old-fashioned, so May I is the standard polite request. Might also offers a suggestion tactfully: You might try the side entrance.
See also can / may; modal verbs.
Examples
Use Rosa may have misread the schedule; she has not answered her phone. (Whether she misread it is still unknown, so may have fits.)
Use Jordan warned that the forecast might change by evening. (After the past-tense verb warned, might places the possibility at the time Jordan spoke.)
Avoid If the alarm had not sounded, the fire may have spread to the next building. (May have is the error: the fire did not spread, so the outcome is contrary to fact and takes might have. Write If the alarm had not sounded, the fire might have spread to the next building.)