In order to begin to understand complexity, we need to discover what complexity is not. Complexity is the region between the polar opposites, *Determinate* and *Probabilistic*. Something determinate can be predicted with certainty. Something probabilistic can be predicted within some statistical bounds (like the flipping of a coin). While complexity includes elements of both of these, it is neither determinate nor probabilistic.

## Determinate

From the New Oxford American Dictionary:

determinate: having exact and discernible limits or form

:the phrase has lost any determinate meaning. determinable: 1 able to be firmly decided or definitely ascertained:a readily determinable market value.

If something is determinate, you know exactly what its value will be given the inputs. Take a multiplication problem as an example:

A multiplication problem has 2 inputs. If you know those inputs, you know the output. Every single time that 2 and 3 are inputted into a multiplication problem, the process yields a 6. There are no instances of 2 multiplied by 3 that result in anything other than 6.

Since you can determine the output given the input, a process like multiplication is determinate. This process is completely determined by the inputs, and the only way you can change the output is by changing the input.

It is possible to create complex results using iterated determinate functions.

## Probabilistic

From the New Oxford American Dictionary:

based on or adapted to a theory of probability; subject to or involving chance variation

:the main approaches are either rule-based or probabilistic.

If something is probabilistic, then each of the possible outcomes has a chance of being the result of a given process. Take the example of flipping a coin:

There are two outcomes, 1) Heads or 2) Tails. Each outcome has a `p(success)=0.5`

or 50%. So, each time you flip the coin there is no way to determine the outcome. In the long-run (1000 flips, for example) and out to an infinite number of trials, however, each outcome will happen just as often as the other. Important about this process is that it requires no inputs – that means that no matter what you input into the process, it will have absolutely no effect on the result.

Here is our coin process redrawn with inputs that include the force with which you flip the coin, the planet where the coin is flipped, and the material of the coin. None of these inputs will change the fact that each side of the coin has a 50% chance to be the output.

If you cannot give the process inputs (or if the process is independent of the inputs), then you have no control over the outcome.

### Randomization

made, done, happening, or chosen without method or conscious decision : a random sample of 100 households.

• Statistics governed by or involving equal chances for each item.

A special case of a probabilistic function is where the outcome is completely random. In the case of the flipped coin, the possible outcome of a flip is either heads or tails. On an evenly balanced coin, the `p(heads) = p(tails)`

. Another way to say this is that the outcome is uniformly distributed, and that each outcome has an equal chance as the other. Since all outcomes are equal, the outcome is completely random.

Another example is a 6-sided die. The outcomes are such that

`p(1) = p(2) = p(3) = p(4) = p(5) = p(6)`

for a balanced die. This is a uniformly distributed probability curve:

## Compare and Contrast

Lets look at the similarities and differences between determinate and probabilistic processes.

### Similarities

Both have a degree of certainty about them: a determinate process will always produce the same output for a given input, and a probabilistic function will (in the long run) produce results that match the likelihood of each outcome. (What if I had a weighted coin that made it 60% likely to land on heads? In the long-run, you can bet money on heads every time and be certain to come out ahead. Even though each toss of the coin is independent, the long-run results are almost certain.)

### Differences

The outcome of a purely determinate process is completely determined by it’s inputs, but the outcome of a purely probabilistic process is completely independent of it’s inputs.

## Complexity is Between Determinate and Probabilistic

Something that is complex is neither deterministic or probabilistic; you can’t easily determine the outcome of something complex. Complex things lie somewhere between the poles of the “Probabilistic-Deterministic Continuum”:

Some outcomes might be more probable than others, but each time you enter the exact same inputs you might get something different or something previously unthought of. Complex functions might be more deterministic in nature, more probabilistic in nature, or strike an even balance.

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