Can someone clarify for me the proper usage of каков compared to какой, particularly situations where you consider каков to be the only option that is allowed?
I have seen какой a lot, but about the only place I can recall seeing каков is in some problem books where questions start off with Какова вероятность, что .... I don't have much of a feel for when каков is required.
Russian adjectives come in two forms: full and short. Compare красный / красен, страшный / страшен, подлый / подл.
The full form is actually the short form with addition of the old Slavic 3-rd person pronouns *jь, *ja which most probably served as a definite article, like -ът, -та in modern Bulgarian.
So actually, we speak about two pairs of pronominal adjectives here: как / какой and каков / каковой.
The two pairs were pretty synonymous in CS but in modern Russian, the first word (как) has lost its adjectival meaning, and the fourth (каковой) has lost interrogative aspect. So the modern language only pairs the words какой / каков as interrogative pronominal adjectives.
What we need to know from all this is that it's the full and the short form of the same interrogative pronominal adjective.
Now, to the usage differences. Russian does not use the short form as heavily as other Slavic languages do, and in neutral speech its usage is limited to stressing rhematic role of the subject.
Compare: это белый дом (this is a white house) // этот дом бел (as for this house, it is white).
The first sentence (это белый дом) does not necessarily imply that we know the topic of the sentence beforehand, and can be a first sentence in a conversation. Белый may define the topic here as well.
The second sentence (этот дом бел) implies that the house is already the topic, and бел here is strongly rhematic.
The short form may imply firmness of the statement (дом не чёрен, он бел // the house is not black, it is white), suprise (почему тарелка пуста) and the similar aspects (when the rhematic role of the adjective is important).
One more example is F. Scott Fitzgerald's novel "Tender Is the Night", whose title is translated into Russian as Ночь нежна (not Нежная ночь). As you can see from the English original, "tender" is clearly the comment in this sentence and the short form is the right form to convey this meaning.
Note that in neutral language, the short form is rarely used in sentences with an explicit verb: горит белый дом but not **горит бел дом. The latter form may still be used in poetic language but it would be considered a slavonicism. What constitutes the rheme here is that the house is burning, not it's being white.
So answering you question: какой and каков are interrogative pronominal adjectives in full and short forms, appropriately, which require an adjectival answer in respective form.
For the following sentence:
Какова вероятность, что …
a grammatically acceptable answer would be:
Вероятность мала.
or
Вероятность велика.
which implies that we know the topic of the question (probability) and the answer we need to give would just add the comment (how high is the probability).
This is exactly when the short form is used.
