May versus might in a sentence?
Ia. Some people follow the rule that may can be considered a greater possibility than might.
“They both indicate that something is possible, but something that may
happen is more likely than something that might happen.”
Quick and Dirty Tips
“’May’ simply states the possibility or likelihood, while ‘might’
emphasizes the conditional nature of the possibility, introducing a
greater level of uncertainty.”
After Deadline NYT
Ib. However, in practice most people use may and might interchangeably in most cases.
“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.”
English Grammar secrets
IIa. May can be considered permission
Example: “Mother may I visit my friend?
II b. It is recommended to prefer might not over may not for clarity because may can be considered "permission"
According to Garner’s Modern American Usage (a bible for all grammar
nuts like me), you should never use “may” in a negative hypothetical
because the reader could read it to mean the person “does not have
permission.” For example, saying “I may not go to the store” could be
misread as “I am not allowed to go to the store.” In this case, you
always use “might”: I might not go to the store.
III. It is recommended to prefer might have over may have for a hypothetical that didn’t happen
Example: when using “may have” and “might have” you can use either if the truth of a situation is still unknown (i.e. "I think your comment may have / might have offended some people"), but if the truth of the situation is known and you are speaking about a hypothetical alternative that did not occur, it’s clearer and more common to use “might have” (i.e. "If Brazil’s best player hadn’t been hurt, they might have won the World Cup").
IVa. Historically different tenses:
In historical practice for the usage of these words some people used
“may” exclusively for present tense and “might” exclusively for past
IVb. Modal usage most commmon today
Today the words may and might are used as modals like would, should and could, meaning that they are generally used without regard to their historical tense and the modal used should reflect a slightly different meaning.
IVc. Even today, especially for some situations involving past tense, many people feel might is still not always interchangeable with may in normal usage.
For instance, because it sounds awkward to mix past tense and present tense, many people including myself would probably prefer “after I read that book, I knew I might visit London one day” to using 'may' (but if someone did use 'may' I could live with it, just wouldn't prefer it).
On the other hand, I find “Last night the weatherman said it may rain today” acceptable, especially because whether it will rain is something that is still an unknown.