I distinctly remember the professor in the undergrad introductory systems & control course saying that "when weather forecasters say there's a 50% chance of precipitation, they are conveying no information".
At the time (freshman) I didn't know Shannon or Kolmogorov but it struck me as a strange comment, after all, if true, why bother reporting the numbers in the first place?
More recently, in Itzhak Gilboa's Theory of Decision under Uncertainty I found the comment, p.25:
First, if we have a random variable X on [0, 1], and we know nothing about it, we cannot assume that it has a uniform distribution on [0, 1] and pretend that we made a natural choice.
So even though the uniform distribution is the maximum entropy distribution (eg among binomial distributions, {rain,no rain}), doesn't saying 50/50 chance convey more information than not knowing the distribution at all?
I'm 50% sure some will vote to close this question, but note I'm not asking for a quantification of how much more (if any) information, but rather just a binary yes/no, which should be within the realm of mathematics.
It does convey information- you must take conditional probability into account. Suppose that I have some information $I$, and that in general the probability of a certain point having rain is $P(R)$. What the weatherman is giving is $P(R|I)$, the probability of rain given the information. It could be that $P(R)=0.1$, while the $P(I)=0.1$, $P(I|R)=0.5$- then by Bayes rule $P(R|I)=\frac{P(I|R)P(R)}{P(I)}=\frac{0.5*0.1}{0.1}=0.5$, which is rather a lot more information than simply knowing that $P(R)=0.1$.
Put another way, 50% chance of rain gives information regarding the weather. In general, one does not expect a 50% chance of rain, in the absence of additional information one would be led to believe the odds were much lower, it is only in the context of the additional information in the weather that we can deduce a 50% chance of rain- which relationship can be used to deduce some information regarding the weather.