It is difficult to use the term random without creating misunderstandings. I try to avoid using it in discussions like the current one. Dictionary definitions and common notions tend to be simplistic. Some consider it a synonym for unpredictable, but incredibly precise clocks have been based on random phenomena. I think it is better to think of it as a term applied to data associated with probability theory, and not try to find a definition in 20 or less words.

Perhaps it is best to consider it defined by the context of its usage.
The mathematics of statistics and probability theory are applicable to phenomena referred to as random processes. The data resulting from such processes is sometimes called random data. Pseudo random number generators are often used by software which attempts to simulate random processes.
Perhaps the implications of statements like the above should be used as the only definitions of random, in which case knowledge of so called random processes and/or probability theory is required for an understanding of the term random. Formal logic uses terms undefined except by their context in the axioms. Perhaps random should be similarly treated as an undefined term in the discipline of probability theory.

Following is some background on the mathematics of probability.

First, I am fairly sure that probability was first formally studied by Descartes and Pascal in the sixteen hundreds. It was initially developed due to questions about various gambling games. By the late eighteen hundreds, probability theory was about as well developed as it is today.

It is interesting to note that while probability theory was being developed, the universe was considered to be deterministic. The mathematicians and physicists of that era believed that probability theory was valuable only because pertinent deterministic laws had not yet been discovered and/or because it was it was not possible to collect all the data and do the calculations required for obtaining analytical answers to certain questions. By the end of the 19th century, it was generally believed, that it was possible in principal to predict the future of the universe, but that in practice, it could not be done.

While developing probability theory, mathematicians made assumptions relating to the nature of uncaused or random events, while believing that there were no such events. It was generally believed possible in principal to predict the results of individual dice tosses and single plays at the roulette table, but impractical to do so. The laws of probability developed by believers in determinism were quite successful in analyzing gambling games and other phenomena. However, most people believe that craps and roulette are governed by the laws of mechanics, generally believed to be deterministic.

By about 1920-1930, various quantum processes were known to produce data that matched the predictions of probability theory very precisely. Furthermore, the Uncertainty Principle was accepted by most physicists. The implications are profound. Knowledgeable people still consider craps and roulette to be governed by the laws of mechanics, but no longer feel confident that those laws are deterministic. It is not the mathematics behind the laws that is questioned, it is the physical processes and the applicability of the deterministic mathematics that has been called into question.

Unassailable mathematical logic developed a theory about uncaused or probabilistic processes. The theory was developed by men who believed in determinism, but needed a practical tool for use when applicable deterministic equations were unknown or impractical. It was an abstract discipline. Then physical processes were discovered that matched the mathematics, and were such that one is not compelled to believe cause-effect relationships existed. Dice and roulette are complex processes, allowing one to imagine some complex underlying deterministic causes. Radioactive decay and various quantum processes do not seem complex. If there is some deterministic cause, it should be obvious or at least imaginable.

The data associated with processes like radioactive decay and the implications of the Uncertainty Principle compel me to believe in a non deterministic universe.

Physicists are aware that quantum precesses underlie all the all the phenomena of the world of our senses. If the quantum processes are not deterministic, then the world of our senses cannot be deterministic. Radioactive decay is the only phenomenon which some non-physicists recognize as being caused by underlying quantum processes, so it is a good phenomenon for us to consider.

Have you ever wondered why the half life is used to describe the rate of radioactive decay? Why not the whole life? The answer is that the half life can be determined very precisely for many radioactive materials, but the whole life can only be approximately measured or predicted.

If the half life of a material is one year and you have 16 grams of it, in one year 8 grams will have decayed, leaving 8 grams. In two years 4 grams will remain, and in 3 years there will be two grams. Similar measurements can be made for the 10% life, the quarter life, et cetera. The half life was chosen as a convenient but arbitrary way of specifying the rate. The decay rate is so precise that some of the most accurate clocks ever built were based on radioactive decay.

Radioactive decay could be modeled by assuming that god or the laws of physics have assigned a coin to each atom of a radioactive material. During the half life, each coin is flipped. If the coin lands heads, the corresponding atom decays. If it lands tails, the atom survives.

Oddly enough, if you have 16 atoms instead of 16 grams, the decay rate is not so precise. In one half life only 6 or 7 atoms might decay, or perhaps 9 or 10 might decay. It is possible that all 16 decay or none decay, but these possibilities are unlikely. If only one atom remains, there is a probability of ½ that it will last two half lives, 1/4 that it will last 3 half lives, et cetera. So the whole life cannot be precisely specified. In fact, the half life for a few atoms cannot be precisely specified.

Probability theory is some times referred to as the law of large numbers. If a large number of trials are made, the data is expected to follow the laws more precisely, and in practice this happens. For a smaller number of trials, the reality does not match the theory as well. This is exactly how radioactive decay works. It is similar to what happens in a casino. If you make about one hundred bets or less at the craps table, you might be a winner (It happens about 40% of the time), but if you play 6-8 hours a day for several days, making many thousands of bets, the casino will slowly but surely exhaust your bankroll.

A gram of a radioactive material consists of more than 10E21 atoms. With that many atoms, the predictions of probability theory match the decay rates with astonishing precision. As expected. For100 atoms or less, the probabilities are not accurate predictors of the number of decaying atoms, also as expected by the theory.

The above compels me to believe that radioactive decay is a probabilistic process rather than a deterministic one. This view also seems consistent with the uncertainty principle.

All quantum world processes seem to be similarly probabilistic. Since the world of our senses is based on quantum world phenomena, it seems reasonable to believe that the universe in general is probabilistic rather than deterministic. I consider it silly to believe otherwise.

What could be discovered in the future to contraindicate the probabilistic view? The nature of radioactive decay was well known before science was aware of the structure of the nucleus. When protons and neutrons were discovered as underlying building blocks, the probabilistic data did not suddenly seem to be deterministic. Science still had no deterministic explanation for why a nucleus decayed. Similarly, QCD theory with its quarks and gluons neither changed the probabilistic nature of the data nor did it provide a deterministic explanation for radioactive decay.

Suppose science discovers structure underlying the quarks and gluons and/or replaces QCD with a better theory? Such discoveries cannot change the probabilistic nature of the data. So long as the data is probabilistic, how can it be rational to believe that the phenomenon will ever be discovered to be deterministic?