The use of "random" to mean "arbitrary", "unidentified", "unknown", etc

Solution 1:

The definition of arbitrary does include a link with random:

existing or coming about seemingly at random or by chance or as a capricious and unreasonable act of will.

It is reflected by the usage "at random" (by chance), which is valid.
The problem is, random is with other interpretations of the word "random":

Less widely accepted are a couple of slangy uses of the word, mostly by young people.

• In the first, “random” means “unknown,” “unidentified” as in “some random guy told me at the party that I reminded him of his old girlfriend.”
• The other is to use random to mean “weird,” “strange,” as in “The party at Jessica’s was so random, not what I was expecting at all!”
  Evidently in this expression randomness is being narrowed down to unlikelihood and that is in turn being connected with strangeness, though randomness in real life is usually quite ordinary and boring.

Use of either of these two expressions in formal speech or writing is likely to annoy or confuse your audience.

Solution 2:

I typically use arbitrary when discussing planning or decision-making, or with respect to preference:

"Your choice of shade of blue for that chair seems arbitrary"

and random in cases where selection or ordering are the relevant aspect, or in mathematical or computer programming contexts:

"I'll shuffle these cards to put them in a random order"

That said, I don't think it's necessarily incorrect to use them interchangeably in some cases; it depends on the context:

"I don't find the placement of these tchotchkes as pleasing when they are strewn about in such an arbitrary fashion"

or

"I don't find the placement of these tchotchkes as pleasing when they are strewn about in such a random fashion"

Even in this case, for myself, the meaning of those two sentences has a slightly different color, but in essense they are interchangeable.

Solution 3:

The difference becomes clearer when you consider the meaning of the word arbitrary.

Arbitrary, means something reached through the process of arbitration. An arbitration is the process of reaching a decision between two or more opinions or parties.

While a random sequence has no defined structure or pattern, an arbitrary one does. Random is the result of pure chance – like when we throw a dice – while arbitrary is a result of choice, deliberation, and often negotiations, sometimes extensive ones.

A random sequence of numbers is never wrong, while an arbitrary one can be. Whichever opinion underpins an arbitrary sequence or system, you will need to follow that sequence in order to get it right – whether you agree with the underpinning opinions or not.

A random sequence of numbers could be made into an arbitrary one – by choosing a sequence of random numbers, and making them “right” and thereby any other combination “wrong”. This is exactly what we do when we create a password, or a combination of numbers for opening a safe. They may start as random, but once we have chosen them, they become arbitrary.