What do we mean by randomness in probability?

We hear about random events all the time, but often have never explored the proper definition for what it means for something to be random

Random processes:

That is something I have often pondered and read about. Diving into the question and exploring what is randomness will be worth the time. The answer to this question is how variables change the process of the outcome of an event.

If the causal factors influencing the event follow a type of random process then the event could be deemed stochastic.

For example, think of physical phenomena that are noisy data, such as atmospheric noise, which can be a causal seed used to generate a random number.

The question is, can atmospheric noise be predicted? If you can predict patterns in atmospheric noise then you can potentially predict the “random number” that is generated.

Maybe, but data supports that natural atmospheric processes are approximately white noise, hence random in fashion. Think about how many possible tens or hundreds of variables contribute to small differences in atmospheric noise given how large and complex the earth’s climate is and the impact humans have on it in their day-to-day operations.

It must be that we don’t know enough of the variables that contribute to differences in atmospheric noise levels, therefore we deem the process as random.

There are several methods of generating random numbers, and, for security reasons, people tend to look for processes in the world that are generally quite random and use those as seeds in generating random numbers. Another example can be the rate of radioactive decay.

The caveats are that even though you might think it is intuitive these processes are quite random; sometimes with large enough data-sets, there are some patterns that can be found, including anomalies.

It turns into a philosophical and exciting mathematical question to quantify how random a process is. I believe the metrics would be up for discussion by the academic literature and community.

Physical objects:

Let’s talk about another example, dice.

I watched a 2 part video today on Numberphile about a mathematician who’s worked on probability theory and dice. Now we are drifting to a tangible object to discuss random events, but keep in mind the game changes in physical objects that we have control over.

I would recommend watching this video as Dr. Diaconis goes on to explain his research on analyzing what is fairness when it comes to dice.

It might seem obvious but it is also something we often take for granted for, an overlooked detail because it isn’t important enough.

See those dice with holes engraved for the dots on each side? Ever thought about how that changes the mass of the dice on each face? Think about how a factor like that affects the outcome of a “random” event we deem dice rolls to be.

It gets more fascinating when discussing objects that have more than 6 faces, beyond the traditional cube.

When it comes to physical objects, the probability of randomness becomes a physics and mathematical problem. The velocity on revolutions, angular momentum, kinetic energy, number of sides and vertices, etc.

Dr. Diaconis goes on in another video to discuss that there are 12 parameters that affect coin flip outcomes. Imagine 12 variables explaining the outcome of a coin flip, a 12-dimensional problem.

I discussed randomness in the sense of data-generating processes that have massive complexity in the real world and how they can be deemed random because we don’t know and/or understand the parameters behind them, such as atmospheric noise.

Now it helps to compare this against a physical object closer to home that we are all familiar with which is why I mentioned die and referenced these Numberphile videos.

If it ever helps, the mental model I use to answer the question you asked is to simply imagine a regression of variables to explain the dependent variable — the event you are hypothesizing to be completely random — and ponder how that model can be constructed with explainability, accuracy, and predictability.