[twoodfin]

If I’m trying to guess a 9-letter English word, and test whether the first letter is “x”, there are only the same two answers: Yes/No.

But “Yes” obviously gives me much more than one bit of the information I need to know the answer.

[Dylan16807]

But that "yes" is so unlikely that your expected/average information is still 1 bit or less.

[twoodfin]

The claim was that one bit was the maximum amount of information you could gain, which is clearly false.

Just to make this unambiguous: If you ask me to guess a number between one and one billion, and by fantastic luck I guess right, your “yes/no” answer obviously gives me more than one bit of information as to the right answer.

[Dylan16807]

> The claim was that one bit was the maximum amount of information you could gain, which is clearly false.

That's not what I see.

https://news.ycombinator.com/item?id=46282007 They have an example that calculates the expected information gained by truth booths and all of the top ones are giving more than one bit. How can this be? It is a yes/no question a max of 1 bit should be possible

https://news.ycombinator.com/item?id=46282343 the expected information (which is the sum of the outcome probability times the outcome information for each of the two possible outcomes) is always less than or equal to one.

The specific comment you replied to had one sentence that didn't say "expected" or "average", but the surrounding sentences and comments give context. The part you objected to was also trying to talk about averages, which makes it not false.

[twoodfin]

If both of these are equally likely, you gain one bit of information, the maximum possible amount. If you already have other information about the situation, you might gain _less_ than one bit on average (because it confirms something you already knew, but doesn't provide any new information), but you can't gain more.

Can’t gain more!

The core confusion is this idea that the answer to a yes/no question can’t provide more than one bit of information, no matter what the question or answer. This is false. The question itself can encode multiple bits of potential information and the answer simply verifies them.

[Dylan16807]

> Can’t gain more!

"you might gain _less_ than one bit on average [...], but you can't gain more."

On. Average.

That's a true statement. Can't gain more than one bit on average.

[twoodfin]

I’m not arguing with that, it’s basic information theory.

One bit, however, is not “the maximum possible amount” you can gain from an oracular answer to a yes/no question. The OP covers exactly this point re: the “Guess Who?” game.