While working on a project involving binary trees as a way to model very simple binary decisions, I began to wonder how many decisions on average does the average person make? A quick Google search for “average daily decisions” led to a number of articles touting a massive 35,000 decisions made everyday by the average person. If you feel skeptical about that number do not worry, you are not alone, but for the purpose of this blog post (which is just a vehicle for some interesting math) it works.
If we assume the simplest case, where each decision has only two options and is hence binary (i.e yes/no, stay/leave, etc…), then by evaluating $2^{35000}$ we can calculate all possible combinations of average daily decisions. This turns out to be quite trivial in Python but the number generated is massive:
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33070135459986386370743434255380996670141988886630
3004530035715655988218319769620709376
This number has $10,537$ digits and represents all possible combinations of average daily (binary) decisions. Because each decision is binary, then the combination of decisions can be represented as a binary number, where the bit-length is the number of decisions (e.g. in this case $35,000$ bits). This means there is a unique number that describes each combination of decisions.
It is easier to see the relationship between binary numbers and combinations of binary decisions when we look at a smaller number of decisions. For example, instead of $35,000$ decisions, we can look at just $4$:
{}
├── 0
│ ├── 0
│ │ ├── 0
│ │ │ ├── 0
│ │ │ └── 1
│ │ └── 1
│ │ ├── 0
│ │ └── 1
│ └── 1
│ ├── 0
│ │ ├── 0
│ │ └── 1
│ └── 1
│ ├── 0
│ └── 1
└── 1
├── 0
│ ├── 0
│ │ ├── 0
│ │ └── 1
│ └── 1
│ ├── 0
│ └── 1
└── 1
├── 0
│ ├── 0
│ └── 1
└── 1
├── 0
└── 1
The binary tree shown above depicts all the possible binary numbers of bit-length $4$. At the root of the tree is the empty set: {}. This represents the initial state where no decisions have been made yet. As we begin to move down the tree, each branch represents a choice (in this case either $0$ or $1$). If we follow the path to the end of the tree, making our binary decisions as we go, we eventually arrive at a unique combination: $0000$, $0001$, $0010$, $0011$, etc… So each path through the tree represents a unique combination of decisions. That combination forms a unique binary number with values from $[0, 2^4 - 1]$.
As stated earlier, assuming the simplest case of two options to choose from (i.e. binary), then each unique combination of $35,000$ decisions has a unique number in the range $[0, 2^{35000} - 1]$. And there are $2^{35000}$ or $10,537$ of these unique numbers, and hence unique paths. So the number $2$ represents some path in this giant hypothetical binary tree. So does the number $2^{35000} - 1 = 10,536$.
Binary numbers have a powerful role to play in a wide variety of applications. Whenever the problem involves some binary component (e.g. binary decisions), you can rest assured that binary numbers have some relevance. The math depicted here, while assuming the ideal situation where each decision is binary, offers us an example of how binary numbers can be used in modeling combination problems in a very compact, and elegant way.