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In common parlance, random and arbitrary are often used interchangeably. A quick check of on-line dictionaries confirms that the semantic overlap is well established in spite of the different origins of the two words.

The fledgling proof-writers need to be made aware that this is not the case in math, with random being used when probabilities are involved. On the other hand, "Let $x$ be an arbitrary integer; then $P(x)$ holds" translates $\forall x \in \mathbb{Z} \,.\, P(x)$ into English.

Next, it would probably help the aforementioned fledglings if they were shown why the distinction is useful. One practical reason is simplicity. If one deals with an arbitrary integer $x$, all that is assumed is that $x \in \mathbb{Z}$. Could $x = 25$ be true? Of course! Could $x = 25$ be false? Certainly!

If, however, $x$ is a randomly chosen integer, not much may be said without knowing the distribution from which $x$ was drawn. The probability of $x = 25$ may be greater than $0$ if the distribution is not uniform (as it must be if the sample space is countable). Besides, as you may well know, zero probability doesn't mean impossible. By avoiding the use of random all these issues are sidestepped.

In more advanced courses, students will be able to appreciate more reasons for keeping random and arbitrary, as well as probabilistic and nondeterministic, distinct. But the example above should be enough to get them started. At any rate, in framing my feedback to students at their first attempts with proofs, I'd assume that they had the right concept in mind, but didn't pick the correct mathematical term to express it.