What's the syntactic explanation in "Mistakes are likely to happen":

Solution 1:

As pointed out in comments, (be) likely is a Raising predicate. That means that the noun phrase subject of be likely with a following infinitive complement is not really the subject of likely, but of the following infinitive clause.

It's been "Raised" up from its position as infinitive subject to a role in the clause the infinitive is part of. Since likely is intransitive, it only has a subject, not an object. So the underlying structure is something like

• [[For mistakes to happen] is likely]

where the subject of is likely is the whole infinitive clause for mistakes to happen.

English doesn't like this kind of sentence, with long complex subjects and short verbs occurring at the end. There are a number of transformations (like Extraposition and Subject Raising) that have the effect of moving heavy subjects to the end of the sentence, producing distinctive structures. Both are possible with this sentence:

• It is likely for mistakes to happen. (Extraposition)
• Mistakes are likely to happen. (Subject Raising)

Raised subjects of infinitives don't use for, which otherwise marks the subject of the infinitive, while it's still the subject. The for is optional with some verbs, disallowed with others, and required when the infinitive clause is subject.

• I intended for him to mail the letter.
• I want (for) him to mail the letter.
• For him to mail the letter was the plan.
• *Him to mail the letter was the plan

This is the kind of Subject-Raising called the A configuration, or A-Raising.
There is also B-Raising, as in

• I expect mistakes to happen.

where the Raised subject becomes the object of a transitive upstairs predicate like expect, instead of the subject of an intransitive predicate like likely.

There is a vast literature on Subject-Raising, starting with Postal's book On Raising. Here's a collection of handouts and exercises with answers on the contrast between Raising predicates and their converse, which are called Equi predicates after the rule they govern.