A word to describe the expected amount of information gained from learning the answer to a given question

Solution 1:

You mentioned the technical term

entropy

which means, in the information theory sense, the number of bits of information in a message, or likewise the thing you are asking for. It is however pretty technical, and also has its provably equivalent but equally obscure interpretation in thermodynamics. As such, replacing the informal phrase 'amount of information' with entropy will be only understood under very technical contexts.

You may be looking for an informal synonym, one that can replace 'entropy' to get the same idea, analogous to your example of 'salt' for 'sodium chloride'. As 'entropy' is a latter day technical invention for a new concept (by Rudolf Clausius in 1854), there is no existing informal older term.

So, there is no single informal word for entropy.

That's a hard thing to prove definitively (as opposed to suggesting a term that every one can see and judge yes or no). But the timing of the term is some justification that it is unlikely.

Asking for a single word for a complex concept is expecting a lot of any language. English and more likely its fans may have spoiled you by popularizing all the ineffable situations that turn out to have single words for them (eg 'justice', 'game', 'ineffable')

However, there may well be near enough synonyms or synonymous phrases for the concept. The title 'amount of information' is sufficient or even simply

'information'.

Solution 2:

One question can be a better discriminator than another.

A characteristic which enables people or things to be distinguished from one another.

[Oxford]

Although the question does not contain the sought-after information, one question has the ability to elicit more information than another. You will find references in the psychological testing domain to high-information yielding questions as good discriminators. For instance, if a particular math problem covaries meaningfully with IQ, that problem is said to be a good discriminator for IQ.

If I'm trying to find a criminal suspect who is an American male, the question "Is he over 7 feet" is a poor discriminator, because it provides very little information for the answer "no". An If I use the median of the population of 5 feet, 9.5 inches, the question is guaranteed to eliminate half the population, which is the maximum one can guarantee, regardless of the answer.