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Under the American Heritage Dictionary's usage note for the verb "might" it says:

Usage Note: May or might? In many situations, the choice between these two verbs can be clarified by remembering that might is the past tense form of may, and that in English, a past tense form is used to refer not just to events that occurred in the past (She left yesterday), but to hypothetical, counterfactual, or remotely possible situations (If you left now, you'd get there on time.)

It's emphasizing that in the following:

If you left now, you'd get there on time.

We use the past tense "left" for a hypothetical, counterfactual, or remotely possible situation.

If I don't use the past tense "left" and instead use the present tense:

If you leave now, you'd get there on time.

I feel that this has the same meaning. Am I wrong?

Further:

Thus, the past tense form might is appropriate in this sentence about a future event that is a remote possibility: ‘If I won the lottery, I might buy a yacht’, which contrasts with the present-tense version that indicates an open possibility: ‘If I win the lottery, I may buy a yacht’.

The first example it describes as a "remote possibility", and the second is described as an "open possibility". Is this implying that "might" is used when there is a lower likelihood in the conditional statement? In the following examples:

• If I won the lottery, I might buy a yacht.
  
  


• If I won the lottery, I may buy a yacht.
  
  


• If I win the lottery, I might buy a yacht.
  
  


• If I win the lottery, I may buy a yacht.
  
  

I struggle to see a significant difference in their meanings, if any at all. All of them mean to me something like:

• If in the future I win the lottery, I may/might by a yacht.

I don't consciously see any difference between "remote possibility" or "open possibility", or any distinction in probability. Am I wrong?

It then explains that when using "might" and "may" to denote past counter-factual or conditional situations their usage panel generally disapproves of "may" and favors "might":

Since about the 1960s, however, people have started using may have where might have would be expected (as in, ‘If he hadn't tripped, he may have won the race’). Although this usage is common in casual speech, it is considered unacceptable by the majority of the Usage Panel. In our 2012 survey, 97 percent of the Usage Panelists found the sentence ‘If John Lennon had not been shot, the Beatles might have gotten back together’ acceptable. Only a third of the Panel (32 percent) approved of the same sentence with may have replacing might have.

So I understand its disapproval by their usage panel. Next, it says another thing that I also don't understand:

Using may have for a past counterfactual situation instead of might have is not only frowned upon by the Panel but can also lead to confusion, since may have is best suited for a different kind of situation: present uncertainty about a past situation. Keeping the two forms distinct reduces ambiguity. ‘He may have drowned’, for example, is best used to mean that it is unknown whether the man drowned, not that the man narrowly escaped drowning.

Again, as far as I can tell both:

• He may have drowned.
  
  or
  
  


• He might have drowned.
  
  

In my opinion, could be read as:

• He may have drowned, he may be dead.
  
  or
  
  


• He might have drowned, he may be dead.
  
  or
  
  


• He may have drowned, he's lucky to be alive.
  
  or
  
  


• He might have drowned, he's lucky to be alive.
  
  

The very last sentence I would say is probably one where "might" sounds a bit strange, and "may" fits better. But the main point is that using "may" or "might" in the bare sentence:

• He may/might have drowned.
  
  

To me doesn't necessarily distinguish between two meanings, nor do I think it suggests it (by itself at least).

Later in the explanation, it says that it's actually been suggested that "might" carries a lesser probability than "may", but I don't personally see this, at least in the examples I've seen.

So how wrong am I? I'm completely blind to nearly all of the distinctions drawn in the dictionary's usage note. Is it that I don't understand the real nuances, or is it that largely they've become interchangeable (a point it also mentioned in the usage notes).

When may and might are used to indicate possibility or probability, as in ‘He may lose his job’ or ‘We might go on vacation next year’, the two words are used almost interchangeably.

Addition:

It seems some (no idea what proportion) make the distinction between might to mean "might have happened (but we don't know) vs may to mean "may have happened (but didn't). This has been mentioned by at least one user who personally makes the distinction themself. I've checked Random House Kernerman Webster's College Dictionary and the entries are:

2.(used to express tentative possibility):
She might have called while you were out.

3.(used to express an unrealized possibility):

He might have been killed!

(2) is described as "tentative possibility", that is, "she might have called while you were out (but we don't know)", and (3) is described as "unrealized possibility", that is, "He might have been killed! (but wasn't)". It does distinguish between what it calls "tentative possibility" and "unrealized possibility", much as AHD's usage note does. If I understand correctly dictionaries usually list their definitions from most to least common, so it's interesting that it hasn't listed them in the opposite order: (3) and (2). I think this is a distinction I've just lost, or never had in the first place. However I like distinctions in words wherever they may be, so I'm happy to learn this and put it into practice, while of course not being a pedant and insisting that it's the "correct" usage of might and may. I'd also like to know if others make this distinction. Maybe you can let me know through a comment.

It may have to do with different dialects.

Dialects A and B are differentiated by reference to constructions like

[1] I thought it might rain before we got home.

[2] I thought it may rain before we got home.

In the older Dialect A [2] is ungrammatical (just like * I thought I can finish the book before I got home): [1], with "might" is required. In Dialect B, [2] is possible as well as [1]. In Dialect A, "might" is undoubtedly the preterite counterpart of "may", just as "could" is of "can" because it is the form required in backshift. In Dialect B there's no basis for retaining (from earlier stages of the language) the analysis of "might" as the preterite of "may": it must be a distinct lexeme.

One factor facilitating this linguistic change is that "might" even in Dialect A is hardly used in the primary sense of the preterite, to indicate past time: we usually say "was/were allowed" rather than "might" for past time permission, e.g. He told me I/we might go.

(Incidentally I have recently noticed that in Frank Palmer's "Mood and Modality" (2nd edn) he says on p. 14 that "might" can be used for past time and then on p.100 that it can't!).

In the grammar books I own, authors will tell students that using the simple past tense in conditional constructions suggest hesitancy and its happening an unlikelihood.

Compare

1. If I am elected… (said by a candidate actually running for office)


2. If I were elected… (said by someone who is not running, and therefore, cannot
   be elected)

Likewise for “left” and “leave”

3. If you leave now,… (there's a strong possibility you can make it on time)


4. If you left now,… (but you're not ready and I suspect you will not make it on time; you are taking too long, you're only half-dressed, you'll be caught in traffic in 10 minutes' time, etc.)

So, some grammars like to suggest there is a subtle (in my view imperceptible) difference between may and might

5. If enough people go out and vote, he may win.
   
   (there's a chance of his winning the election)




6. If enough people go out and vote, he might win.
   
   (There is probably another impediment, maybe the elections are rigged, maybe he doesn't have enough support from the constituents, maybe the polls are low, maybe we know he is hiding something in the closet, etc…)

From A Practical English Grammar by A.J.Thomson and A.V.Martinet (Fourth Edition 1986; Ninth Impression 1990)

Possibility

132 may/might for possibility

A […]

B  may/might + present infinitive can express possibility in the present or future

 He may/might tell his wife (Perhaps he tells/will tell his wife.)

 He may/might emigrate. (Perhaps he will emigrate.)

 Ann may/might know Tom's address. (Perhaps Ann knows etc.)

Similarly with the continuous infinitive:

 He may/might be waiting at the station. (perhaps he is waiting at the station.)

 He may/might be waiting at the station when we arrive. (Perhaps he will be waiting etc.)

C  […]

D   might must be used in the conditional and when the expression is introduced by a verb in the past tense:

 If you invited him he might come

 I knew we might have to wait at the frontier. He said he might hire a car. (indirect speech)

E […]

133 may/might + perfect infinitive

A  This is used in speculations about past actions:

 He may/might have gone =

 It is possible that he went/has gone or

 Perhaps he went/has gone

might must be used, as shown in 132 D, when the main verb is in a past tense:

  He said/thought that she might have missed the plane.
might, not may, must be used when the uncertainty no longer exists:

  He came home alone. You shouldn't have let him do that; he might have got lost. (But he didn't get lost.)

So in the sentence:

  You shouldn't have drunk the wine: it may/might have been drugged would indicate that we are still uncertain whether it was drugged or not. it might have been drugged could have the same meaning but could also mean that we know it wasn't drugged.

To me, he may have drowned means that we don't know whether or not he has survived, and he might have drowned means that he ran the risk but was saved. A lot of writers nowadays don't observe this distinction, though.

Might is the past tense of may.examples[1] you may attempt the question or you might attempt the question [expressing present or future possibility].Both sentences are correct[2]He may have asked me [I don"t remember whether he asked me or not } and he might have asked me [now it is past,there is no time to ask me] I might reach my destination if i start now or I may reach my destination if I start now. The word 'can' is also used.Both sentences can be used to express possibility. Again the word may is used to give permission to one who seeks permission such as "may I go out of the class? is replied by the teacher "you may go out of the class" but if might is used by a fellow student- you might go out of the class-[ temporarily because the teacher is not any where near the school] also the word may expressing wish or hope 'May his soul rest in peace' and the word might expressing annoyance,you might have told me.Asking for information especially condescendingly 'who might you be? concise oxford dictionary Indian edition[tenth] consulted user 26375