modal verbs - In depth explanation of the difference between "will" and "going to"?

Background

My wife and I take English classes provided by our companies, so we have different teachers. So now it happened that we had the same topic: future.

Both teachers are native English speakers, but I don't place too much faith on that. I'm a native German speaker but I couldn't teach anyone how the grammar of German future works.

The question

We (my wife and I but also our teachers) both agree that "will" and "going to" indicate something to happen in the future. We also agree that the difference is just a matter of probability. However, we have opposite understandings on that probability.

My wife / my wife"s teacher says:

"Probably", "maybe" and "perhaps" are keywords pointing towards to use of "will".

There are no keywords for "going to", but it should be used when something is very certain to happen.

I say / my teacher says:

"Going to" is used in cases with low probability

"Will" is used with high probability.

The latter model fits very well to what I have learned at school. However, my wife"s teacher says that one reason for Germans to have the difficulty is that we learn it wrong at school.

When to use "will" and when to use "going to"? Is it a matter of probability? Is it a matter of stylistics (e.g. will for the single sentence examples and be going to for the multi-sentence examples as pointed out in one of the comments)? Is it a matter of how far into the future (present prediction vs. future prediction)?

What I have tried

Oxford Dictionary

I looked up the definition of "will (modal verb)" in the Oxford Advanced Learners Dictionary, sixth impression and it says:

1(a) (indicating future predictions)

1(b) (indicating present predictions)

2(a) (indicating willingness or unwillingness)

2(b) (indicating requests)

3 (giving an order)

4(a) (describing general truths)

4(b) (describing habits in the present or past)

5 (insistence on the part of the subject)

From these descriptions, my opinion is that "will" is used for higher probability (a general truth is more likely than something else, habits are more likely than non-habits). I also can't find any of the keywords in the examples given in the Oxford dictionary.

I also tried looking up "going to", but it's neither present under "going to" nor could I find it under "go".

Inlingua textbook

Second, I looked it up in my English textbook, which is Inlingua English 2 Step 1. Unit 7 is about "Planning" and there is an exercise 7.3 about decisions. While I would consider a decision and plans as something with high probability (because I can influence it), all the sentences start with "going to", which contradicts my understanding. Unfortunately, the Inlingua book is quite crap, because it seldom gives definitions and bases everything on examples only.

Third, there is Unit 8 "Predictions" in the same textbook. Under 8.3 there is one of the rare cases of a definition:

We use "will" and "going to" to discuss forecasts that are quite certain.

This finally confuses me even more, since there is almost no difference any more.

English Page

I also had a look at English Page / Simple future. It says

USE 1 "Will" to Express a Voluntary Action

USE 2 "Will" to Express a Promise

However, I don't see much of a difference between those in the examples. If someone says

I will translate the email, so Mr. Smith can read it.

is that a promise, a voluntary action or maybe both? I just don't get it.

The site also says

USE 3 "Be going to" to Express a Plan

It does not matter whether the plan is realistic or not.

If the plan is not realistic, that would be improbable.

Also, what is the difference between a plan ("going to") and a promise ("will")? What if I promise someone to follow the plan?

Is "I will travel to Switzerland" a plan (e.g. I have made a plan how to travel there), a promise (because I talk to my girlfried living in Switzerland) or a voluntary action (which it also is)? So use "will" or "going to" now?

USE 4 "Will" or "Be Going to" to Express a Prediction

Predictions are guesses about what might happen in the future.

Derived from "might", this sounds both suitable for improbable cases.

Answer

BE + going to - Lindsay is going to fly to New York next week.

Forms with BE + going to possibly originated in such utterances as:

1. We are going to meet Andrea at the cinema,

uttered when we were literally going, i.e. on the way, to the meeting. At the moment of speaking there was present evidence of the future meeting. This use has become extended to embrace any action for which there is present evidence – things do not have to be literally moving. Consider now these two utterances:

2. Look at those black clouds. It's going to rain.
3. Luke is going to see Bob Dylan in concert next year

In [2] the present evidence is clear – the black clouds. In [3], the present evidence may be the tickets for the concert that the speaker has seen on Luke’s desk, or it may simply be the knowledge in the speaker's mind that s/he has somehow acquired.

Modal (will) - Lindsay will fly to New York next week.

Will is a modal and, like the other modals, has two core meanings. The two core meanings for most modal are:

(a) the 'extrinsic' meaning, referring to the probability of the event/state (b) the 'intrinsic' meaning, reflecting such concepts as: ability, necessity, obligation, necessity, permission, possibility, volition, etc.

The extrinsic meaning of will is exemplified in:

4. Emma left three hours ago, so she will be in Manchester by now.
5. There will be hotels on the moon within the next 50 years.
6. The afternoon will be bright and sunny, though there may be rain in the north.

In all three examples, the speaker suggests 100% probability, i.e. absolute certainty. (MAY would imply possibility, MUST logical certainty, to take examples of two other modals). Note that while certainty in [5] and [6] is about the future, in [4] it is about the present. It is the absolute certainty, in the minds of speaker/writer and listener/reader, that can give the impression that forms using ‘the will future’ are some way of presenting ‘the future as fact’. Some writers therefore call this form ‘the Future Simple’. Weather forecasters, writers of business/scientific reports, deliverers of presentations, etc, frequently use will, and learners who encounter English more through reading native writers than hearing native speakers informally may assume that it is a 'neutral' or 'formal' future. In fact the particular native writer or speaker is simply opting to stress certainty rather than arrangement, plan or present evidence.

The intrinsic meaning of will is exemplified in:

7. I'll carry your bag for you.
8. Will you drive me to the airport, please?
9. Jed will leave his mobile switched on in meetings. It's so annoying when it rings.

These examples show what we might loosely call volition, the willingness or determination of the subject of the modal to carry out the action. Note that [9] is not about the future, and in [7] and [8] the futurity is incidental. It is context rather than words which gives the meaning.