Final answer:
The statement is true; when the probabilities of event outcomes are closer to 1/N, the average information is less because it implies all outcomes are equally likely, reducing uncertainty and entropy.
Step-by-step explanation:
The question you have asked relates to the concept of probability and information theory, which are parts of Mathematics. Specifically, you're referring to the scenario where the probabilities of different outcomes are close to 1/N, where N represents the total number of outcomes. The statement 'When event outcome probabilities are nearer to 1/N where N is the total number of outcomes, average information is less?' is indeed true.
Information theory suggests that the amount of information gained from an event is inversely related to the probability of that event occurring. If an event is very likely then little information is gained when it occurs, because it was already expected. Conversely, if an event is unlikely, more information is gained when it occurs since it was not expected. This is quantified using the concept of entropy, which is a measure of the uncertainty or the average amount of information produced by a stochastic source of data.
When probabilities are close to 1/N, this implies that all outcomes are equally likely, which is the case with a fair die or a fair coin as you mentioned. This situation has the least amount of uncertainty and hence, the entropy or average information is minimized. As probabilities deviate from 1/N, some outcomes become more likely than others, increasing the uncertainty and the average amount of information one would gain from observing the event's outcome.
Understanding how this relationship works is important in fields like statistics, where theoretical and empirical probabilities may differ. For instance, if empirical data shows that outcomes are not equally likely, as in a distribution where more people require less time than others, this indicates an uneven distribution of event probabilities, leading to more information content than in a uniform distribution.