May, might, can, and could (possibility 2)

A. Compare these sentences:

I'll write the date of the meeting in my diary, otherwise, I may/might/could forget it.

(= talking about the present or future possibilities)

Jenny's late. She may/might/could have forgotten about the meeting. (= talking about the past

possibilities)

We use MAY/MIGHT/COULD (not 'can') + HAVE + PAST PARTICIPLE to say it is possible that something happened in the past:

I thought I saw Tom in town, but I may/might could have been wrong.

'Where's Barbara's camera?' 'She may/might/could have taken it with her.'

We use MIGHT/COULD (not 'may' or 'can') + HAVE + PAST PARTICIPLE to say that something was possible in the past, but we know that it did not in fact happen:

If I hadn't come along at that moment, Jim might/could have been the one arrested instead

of the real thief.

The plan might/could easily have gone wrong, but in fact, it was a great success.

We use MIGHT (not 'may') + INFINITIVE to talk about what was typically the case in the past:

During the war, the police might arrest you for criticizing the king.

Years ago children might be sent down the mine at the age of six. (passive form)

We can also use COULD + INFINITIVE in examples like this to talk about past abilities. 

For example, 

'During the war, the police could arrest you...' means that the police were legally able to

arrest you.

B. We use MAY/MIGHT(not 'can') + have + past participle to say that by some time in the future, it is possible that something will have happened:

By next Friday I may/might have completed the report.

His maths may/might have improved by the time the exam comes round.

We use MAY/MIGHT (not 'can') + BE + VERB-ing to say it is possible that something is happening now or to talk about a possible future arrangement:

I may/might be working from home today. 

When I go to Israel. I may/might be visiting the Dead Sea, but I'm not sure yet.

C. We use MAY/MIGHT/COULD + well/conceivably/possibly + HAVE + PAST PARTICIPLE (compare Unit17) to say it is likely that something would have happened in the past if circumstances had been different, or to say that by some time in the future it is likely that something will have happened.

(Notice that we don't use 'can well (etc.) + have + past participle):

My tasks may/might/could conceivably have been done if I had had enough time (passive form)

By this time next week, I may/might/could well have left for Israel.