Paul Campbell

The course of one’s life often seems a mystery. Even looking back over the course of the previous year often makes one wonder how so many unexpected twists and turns of “fate” could have brought them to where they are today. In recent years mathematicians have attempted to answer this question with a relatively new field of mathematics called Chaos Theory, or what popular culture and some philosophies refer to as “The Butterfly Effect.” While this Theory is essentially true, it tends to deal with the sum total of humanity on the planet and all living and many non-living systems and how they interact. It doesn’t, in general, deal with the individual and how the effects of one action affect the starting point of the next action.

On an individual basis the path of one’s life can be reduced to a semi-infinite series of Bernoulli Trials. They are semi-infinite rather than infinite because we accept the premise that humans are mortal, and therefore the series will at some point come to an end. We are after all talking about the path of one’s life, and the course of one’s afterlife is beyond the purview of this paper.

A Bernoulli Trial is simply an action which has only two possible outcomes, typically called success and failure respectively, and each possible outcome is equally likely, and the outcome of each successive trial is completely independent of any previous trials. In simplest terms, the classic explanation of a Bernoulli Trial is a coin toss, with only two possible outcomes, heads or tails, and each outcome is as statistically likely as the other, and the outcome of each trial is not effected in any way by any previous coin toss.

From this we can postulate that every single action in life is nothing more than an individual Bernoulli Trial. By way of illustration let’s assume you are going to drive to work. The number of actions between waking up and getting to the car also involves a nearly infinite series of trials, but let’s start with starting the car. It either starts, or it doesn’t. If it starts, we’ll call that success, and if it doesn’t start we’ll call that failure. What either possible outcome is labeled is completely immaterial. Let’s assume it starts.

Now another nearly infinite series of possible outcomes: the car goes in gear or it doesn’t, you back out of the driveway safely or not, and on and on, but the object here is that you either make it to work, or you don’t, there are really only two possible outcomes, and statistically, one is just as likely as the other. While you may have driven to work safely thousands of times, each time the outcome is completely independent of any previous time. In the same way, you can theoretically toss a coin and come up heads a thousand times in a row, but that in no way effects the next toss, which could just as likely be tails.

Let’s now say that you don’t make it to work. Either the car breaks down, or you get in an accident, or you get half way there and just decide you can’t face that job one more day, but for whatever reason, you don’t get there. Chaos Theory would deal with all of the events associated with or leading up to why you didn’t get to work, but the net result on an individual basis remains the same. Now, all of the actions which you would have taken that particular day, at that particular job, never happen to you and are now lost to time forever. Again, Chaos Theory would explain how that would affect not only you but all of the others you would have come into contact with. On an individual basis, none of that matters.

 One’s individual future actions start from the point they are at, and each successive actions remains with only two possible outcomes.

It is obvious from the foregoing illustration that each successive trial doubles the possible positions you can end up at, and given the nearly infinite series of trails to get from any one objective (getting out of bed) to the next objective (getting to work) that even in a relatively short period of time one’s life can take on a completely different, and largely irreversible, path!

This sounds at first blush as though it is not a true Bernoulli Trail, and that the outcome of the previous trial does influence the successive trail, but careful examination shows that the trails are independent, the difference here is that each result puts the individual in a new starting position. Mathematically that position P is a function of the number of trials n from where the individual started.

P = 2ⁿ

This is a basic doubling formula. Sadly, this is an over-simplification because each action, including the decision to take the action, requires some value of time. That time will vary from individual to individual and from decision to decision. Therefore, trying to calculate the time required to get from one position to some subsequent position would be difficult at best. The net result however remains the same over an extended period of time: The position one is in after three or four trials can be vastly different depending on the series of positions the individual lands in from one trial to the next.

Herein lies the reason why “The best laid plans of mice and men often go awry.” Let’s assume that on any given day an individual makes ten decisions, which is a gross underestimate, but it will demonstrate the futility of planning. On New Year’s Day the individual makes a resolution, a declaration of intent, of something they intend to accomplish or a place they want to be in at the end of the year. Assuming ten decisions a day for the ensuing three-hundred and sixty-five days there are 3600 decisions/actions taken throughout the year. The number of possible positions the individual can land in however is:

 P = 2³⁶⁰⁰

This number is calculable, but so astronomically large as to be essentially meaningless. So let’s consider where the individual is in four days:

 P = 2⁴⁰

∴ P = 1,099,511,627,776

In other words, in just four days, taking only ten actions in a day, the individual can potentially end up randomly in any one of approximately 1 Trillion different positions from where they started. On day five they pass 1 Quadrillion, on day six they pass 1 Quintillion, and by the end of the first week of the New Year they are now in any one of 1 Sextillion different possible starting positions. So by the time the day ends on the second Monday of the year our hapless individual is potentially in any one of 1.2 Septillion different possible starting positions.

P = 2⁸⁰

∴ P ≈ 1,208,925,819,614,630,000,000,000

So, yes Virginia, life really does change in a day.

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