The definite article "the" is not used before the superlative mainly in three cases:
When "most" means "extremely". For example, your sentence (1), which is by the way the only one that is correct in your set, means: He is extremely/highly likely to succeed.
When the superlative is not accompanied by a noun, either implicit or explicit. In P.E.Dant's example above:
- Gin is best with a splash of tonic,
there is no noun, either tacit or express, after "best". Actually, "best" there means "ideal" and refers to "gin" itself. However, if somebody asks:
A. What's your favourite drink?
B could answer: In my opinion, gin is the best (drink).
When you use a phrase that indicates the group within which or the place where a certain item shows some feature in its superlative degree (by comparison with all the other items in the group), then you have to use "the" -- that's why your examples (2) and (3) are wrong:
2'. It is the fairest of all methods.
3'. I have read many books. "Against all odds" is the best of all (the books I've read).
Here's an example with a locative phrase:
- Susan is the best student in my class.
3. In some verbal phrases with verbs of liking, "the" is optional:
- Which book do you like (the) most/best?
According to this page, the options with "the" are considered by some to be less grammatical.
In support and as a supplement of what I said under (1) and (2) above (as well as of what @BenKovitz says here: You can even put a in front of a superlative when you intend the superlative to designate only a very high degree of something rather than the one item with the highest degree: The hermitage is a most curious piece of architecture.), we find all of this clearly explained by Quirk below:
The superlative without the definite article may function as a relative superlative (compared with other members of the group) or as an absolute superlative or intensifier (without entailing such comparison).