Learn English – Usage of ‘increase in’ and ‘increase of’

First and foremost, all four of those sentences are in fact grammatically correct. What Google demonstrates is that a.2 and b.1 are really awkward and therefore much less common.

In terms of the implications of the words themselves:

Opportunity often implies that it is an offer being extend or provided to you by a specific entity or person who is providing you with means or privilege otherwise unavailable to you. Sometimes there is no other party offering the opportunity, but there is always the sense of urgency, that it is a limited time deal, something that must be seized or it will be lost. There is generally a positive connotation: an opportunity is a good thing, something to be excited about.

Possibility simple implies that it is something that can happen, either because you decide to make it happen or because circumstances cause it to happen. There is no implicit positive or negative connotation here. There is also no implicit implication of the likelihood of the event happening. If it is possible, it may or may not happen.

In general, opportunities imply possibility, but not the other way around. If you are given the opportunity to go to Norway, then of course it is possible. But if you lived within driving distance of Norway, you wouldn't usually refer to your possibility of driving there at any moment as an "opportunity".

Some examples:

There is a distinct possibility that I may lose my job next week, but I have been given the opportunity to work at an exclusive Swedish design firm, so I'm not too bummed.

I really needed to talk to my professor. He was a busy man, so this was nearly impossible, but I happened to see him walking alone one day, so I seized the opportunity.

Interestingly, this Ngram suggests that in fact the usage of "opportunity to go" and "opportunity of going" have reversed in the last 200 years, with both being approximately equal in use around AD 1890-1900.

Here are some things you can say when your friend is speaking too quietly:

Please speak up!

Please speak more loudly.

You wouldn't say "speak loudly", since that means to speak in a loud voice, not merely to speak more loudly than your friend is currently speaking. You also wouldn't say "increase your voice", since the verb "increase" only makes sense with things that vary quantitatively, and English doesn't normally regard a person's voice as a quantity. (But we do say "Raise your voice" to mean speaking in a way that is both loud and angry.)

Very informally, you can say:

Talk louder!

This is a violation of standard grammar, since it treats "louder" as an adverb. The nonstandardness adds rhetorical effect. "Talk" is also a less formal word than "speak".

Here is how you can ask to change the volume of the loudspeaker:

Please raise the volume.

Please turn up the volume.

Please turn up the music.

Please increase the volume.

You wouldn't say "amplify the volume" because "amplify" means to increase the amplitude of a signal (that is, an oscillation). You can "amplify sound" and "amplify current" and "amplify voltage", but not volume. Amplifying the sound results in increased volume. Also, "amplify" is a somewhat technical term. In English, it doesn't have the meaning of "enlarge" that its cognates have in the Romance languages.

You wouldn't normally mention the loudspeaker explicitly. Usually if you say "the volume", people will know what you mean. "Loudspeaker" is also a somewhat technical term, used only when needed to prevent some unusual ambiguity. Usually people just say "speaker" (even though this also means a person who speaks).

The phrasal verb "turn up" makes your listener think of a knob, which you turn (usually clockwise) in order to increase the volume. People say "turn up the X" to mean "increase X" even if the control doesn't literally turn, like a slider. Informally, "turn up" can be used very loosely and metaphorically, about almost anything.

Here are some informal ways to say the same thing:

Make it louder!

Turn it up!

The pronoun "it" doesn't have an antecedent because you are assuming that your friend will know what you're referring to.