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It seems i can't directly say probabilistic and random are identical .

But this is telling :

random experiment is a probabilistic experiment.

Is there any difference between Random and Probabilistic ?

How are the terms Random and Probabilistic normally used?

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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:

random experiment is a probabilistic experiment.

That still holds, given that random is a special case of probabilistic, so every random experiment is probabilistic but the opposite is not necessarily true.

Note: The common distinction is not between random and probabilistic, but between deterministic and probabilistic.

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    $\begingroup$ This characterization of "random," although perhaps agreeing with some vague colloquial sense of the word, seems somewhat idiosyncratic to me. Could you perhaps cite an authority who defines "random" as implying equiprobable outcomes? $\endgroup$
    – whuber
    Mar 26, 2015 at 15:09
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Here is a good resource: https://plato.stanford.edu/entries/chance-randomness/

There are differences in process and product chance vs randomness. The concept of algorithmic randomness is discussed.

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    $\begingroup$ Welcome to the site. At present this is more of a comment than an answer. You could expand it, perhaps by giving a summary of the information at the link, or we can convert it into a comment for you. $\endgroup$
    – Sycorax
    Oct 27, 2021 at 1:03
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Many mathematicians and statisticians are undisciplined in the use of these words, using the word "random variable" to refer to any quantity to which a probability distribution is assigned.

The number of spades you get in a poker hand from a well shuffled deck is random. The mass of the dwarf planet Pluto may not be random the way a poker hand is, but it may be uncertain in such a way that it is reasonable to express that uncertainty by assigning a probability distribution to the mass.

Suppose you take a random sample of $50$ twenty-one-year-old men from a large population and measure their heights and use the sample average as an estimate of the average height of twenty-one-year-old men in the whole population. After you've sent them home, you draw a new sample of $50$ such men, and do it again. For many purposes in statistics, anything that changes when you get that new sample is "random."

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