Why is the same modal auxiliary, 'may/might' used to ask permission and for uncertainty?

"May I go to the bathroom?" and "I asked if I might go to the bathroom."

The modal auxiliary 'may/might' is used to ask permission.

"He may not have understood your question." and "He might not have understood your question."

The same modal auxiliary is used for uncertainty.

Why?

Solution 1:

According to OED 1, the oldest meaning (attested from the early 9th century) was “to be strong, have power or influence”—a sense still present in the noun might and the adjective mighty.

No later than the end of the 9th century the word was in use to express “objective possibility, opportunity, or absence of prohibitive conditions”.

The permissive sense developed somewhat later, around the year 1000: “To be allowed (to do something) by authority, rule, law, morality, reason, etc.”. This is a fairly obvious extension of the primary sense: to permit someone to do something is to grant them the “power” to do it.

What you describe as the “uncertainty” is attested by 1200. Again, it's a logical step: to say you are able to do something is different from saying you will do it, so may cedes ‘ability’ to can and assumes the sense of “Expressing subjective possibility, i.e. the admissibility of a supposition”.

Solution 2:

Originally, might / may meant having the ability to do so. The noun might as in strength comes from the same root. This differed from the similar rôle of could / can in that originally might referred to power, could to knowledge.

From this origin, we have both the meanings you suggest.

That of uncertainty relates to possibility. It is true that I might have misread your question and be writing a useless answer, it is not true that I might grow wings and fly before I finish this sentence. I have the ability to do one, but not the other, but my saying I might do so does not mean that I will.

That of permission also relates to possibility; someone has the authority to permit or deny something (or perhaps in some circumstances we may cede them that authority before asking their leave, out of politeness), and hence it is they who we must ask if we may / might do something.