I will try harder. Consider entropy.

The first sentence in this comment contains 32 characters; from the point of view of a naïve channel with no compression, that's 256 bits (given none require breaking out of the first bytes of UTF-8).

It did not take 2^256 attempts to construct the first sentence in this comment, because the generation process was not flipping coins per bit.

LLMs also do not emit bits chosen with a [0: 0.5, 1: 0.5] probability distribution.

From a compression point of view, the bits-transmitted-per-bits-in-message ratio can be reduced such that more likely messages use fewer bits than less likely messages. However, this requires the receiver to agree with the sender what the probability distribution over tokens is.

Intelligence is, amongst other things, a compression algorithm. If I can predict your next token, and we both know this, we can agree in advance that you don't need to actually send it.

No single human brain is able to predict the output of an LLM anything like well enough to do that.

In entropy terms: LLMs are noisy sources, their output does contain false statements, yet they add more bits of signal than of noise relative to a human alone.

Or at least, they can add more add more bits of signal than of noise relative to a human alone, but humans who blindly copy-paste the output of an LLM without checking are a pain and add zero value to whatever situation they happen to be in.

For some hypothetical scenario, writing software because I know they can do that, asking an LLM to write some code for you may easily give you 10 kilobits of positive information (code that mostly works), and -30 bits of noise (each bit being one binary decision's worth of incorrect choice by the LLM in what to write, i.e. bugs); if you as a user don't know how to handle the -30 noise that could easily be a totally useless app, but if you can filter out 30 bits of noise, either manually because those 30 bits happen to be your skill set, or even in some cases by prompting it again with the failure mode, then you get to benefit from the 10 kilobits of good stuff that you didn't have before.

In many (but not all) cases, LLMs can fix more than 1 bit of mistakes per follow-up prompt.