Stats textbook are sometimes divided into two sections: 1) Probability and 2) Statistics.
Probability was about permutation, combination, conditional probability. Probability was often explained with dice, coins, colored marbles and other discrete artifacts. Probability is the measure of the likelihood that an event will occur. Although probability can be calculated using statistical models, and probability does not have to be from countable events and a rational number on 0-1, that would be its first meaning. Statistics was about mostly real number measurements, like the t-, z-, W- and U- statistics. A statistic (singular) or sample statistic is a single measure of some attribute of a sample (e.g., its arithmetic mean value).
There is an overlap or gray zone for probability for the discrete distributions including the binomial distribution, the multinomial distribution, and the Poisson distribution, which are still finger counting, i.e., literally countable probable events, for which statistics are parameters and for statistics for continuous distributions that are real number models of probability.
Probability models as a first meaning imply countability, for example, the likelihood of getting exactly 5 heads from 10 coin tosses. Statistical models, as a first meaning contain some statistic. However, one can say that "Marginal probability is a statistic" so that using which phrase is used when depends on what connotation one is making.
Ex Ante means before the event. Ex Post means after the event. In this example, I think this means before and after the event that gives the statistical difference you're testing, respectively.
On the other hand, a priori and a posteriori are terms from philosophy, respectively denoting knowledge that is logically derived, and knowledge that requires empirical evidence. (Wikipedia)
Best Answer
Probabilistic means there is uncertainty in the process where the possible outcomes of some event may or may not have fair (equal) shares. For example: When you throw two fair dice, the probability of getting a sum of the observed top faces greater then 4, is greater than getting a sum of less or equal to 4.
You can think of Random, as a special case of probabilistic where all the possible outcomes of some event in an experiment have equal probabilities of happening. Example: Drawing one card out of a standard 52-card deck gives $1/52$ probability for each of the cards (given no preference or prior information whatsoever about any of the cards). For more information about random chances, check Uniform Distribution. From the definition of uniform distribution: If an experiment is random, the probability of an event is the number of possible outcomes divided by the total number of possible outcomes.
To get back to the text you quoted:
That still holds, given that
randomis a special case ofprobabilistic, so every random experiment is probabilistic but the opposite is not necessarily true.Note: The common distinction is not between
randomandprobabilistic, but betweendeterministicandprobabilistic.