A quote from an old Irishman's Diary in The Irish Times - sadly subscriber-only - about the difference between 'may' and 'might':

"One of the more annoying grammatical errors of our time is a tendency to confuse “may” and “might” when speaking of the past. You’re watching football on TV, for example, and a striker attempts a volley, but instead balloons the shot high over the crossbar. Then the commentator notes that the player had time to control the ball first and adds that, if he had done, “he may have scored”.
"What the commentator means is that he might have scored. Whereas “may” implies that he possibly did score, but we don’t know enough about the incident yet to be sure.
"This even though the ball has just knocked the false teeth out of a pensioner in Row W of the stand behind the goal…"