Pupils may go out if they've finished their homework. Since your homework isn't in fact finished you can't go out.

He can't be serious; no one could believe that. In fact he's an important scientist, so he might be serious after all.

She might get good results if she tries harder. No, it's too late; the exam's next week: she will possibly succeed.

If you want to ask for permission, use "can" or "could".

My parents said "You must tidy your room", so I have to do it.

"Shall" is a little bit politer than "may".

That must be right; your argument is perfectly logical!

If the figure has more than three sides, it must be a triangle.

Since it's not forbidden by the rules, I suppose we can do it.

Someone is knocking at the door. It must be the postman.

Granpa has been working hard in the kitchen all day. He must be exhausted.