Binary Decision Making (and Why it’s Good)

This is the story of how solving a very simple mathematical problem taught me a lot about decision making and my perspective on life. This can even help you make decisions quicker and be stronger in your relationships.

If you know nothing about math or even hate it, don’t worry, this is really easy. First I’ll explain the problem and how it can be solved. Just follow me there, as it is important in order to understand the rest. Then I’ll describe what I have learned out of it about decision making.

Apples in a Bag and How to Solve it

Here is the problem: Given a set X of n elements, how many different subsets of X are there?

Let’s say your set of elements is a shopping bag containing apples. Let’s call the shopping bag X, and the number of apples in the bag n. We assume that each apple is uniquely identifiable. How many different ways are there to distribute these apples in or outside the bag?

– There is only one way to have zero apples in the bag (the bag is empty) and also only one way to have all apples in the bag (they are all in the bag and none is out).
– If we look at all possible ways to have one apple in the bag, then there are n possibilities for that, since we have n apples that could be alone in the bag.
– The number of ways to have two apples in the bag depends on how many different pairs of apples we can form and put into the bag.
– The number of ways to have three apples in the bag depends on how many different groups of three apples we can form and put into the bag.
– And so on.

If we add up all the possibilities for zero apples, one apple, two apples, etc., up to n apples, then we have the total number of different subsets of X.

How do we know how many pairs of apples we can form? Well, for the first apple in the pair, we have n possibilities, since we can pick any of the n apples. For the second apple in the pair, we can pick all apples but the one we already picked, so that is (n – 1) choices. The total number of possible pairs therefore is n * (n – 1), since for each of the n ways to start a pair we have (n – 1) ways to complete it.

But then, we would have some pairs twice. Sets are not ordered, which means that the apples are in no particular order in the bag. Therefore, the subset {apple1, apple3} is the same as the subset {apple3, apple1}. So here we need to take only half of the pairs into account.

With three apples it is the same: we have n possible choices for the first apple, (n – 1) for the second one, and (n – 2) for the third one, which is a total of n * (n – 1) * (n – 2) possibilities. There are, however, six possible ways to order three apples.
1 2 3
1 3 2
2 1 3
2 3 1
3 1 2
3 2 1
Therefore, here we need to take only one sixth of the results.

If you keep doing this for a few more n’s, the pattern that emerges is that the number of subsets with k elements is 1/k! * n!/(n-k)! and after some applying of the binomial theorem, the total number of subsets of X turns out to be 2^n (2 to the power of n). (You don’t need to understand this. What counts is that the result is 2^n. I apologize for not having figured out how to get mathematical notation in wordpress.)

I hope you are confused. I definitely was. I think that is all pretty complicated. Moreover, I was able to prove the result with the binomial theorem, or simply with induction (you don’t need to know what that is), but I did not truly understand it. Demonstrating something is one thing, but understanding why it is that way is another, and I did definitely not understand where this 2^n came from!

A Better Way to Solve the Problem

Fortunately after a while of being grumpy and grumping around, I suddenly got it. It was a matter of focus. My perspective had been ineffective all this time.

Instead of considering all possible subsets, I needed to focus my attention only on one thing: the apples. Or even better, only one apple. No matter which combination we are examining and in which subset it is, each apple can only be either in the bag or not, right? That is a simple yes/no choice. Two options. Let’s call them 0 (no) and 1 (yes).

Now there are n apples. So each subset, which means, each possible way to place these n apples in or outside the bag, can be expressed as a chain of n zeros or ones. The empty bag would be n times a zero. All apples in the bag would be n times the one. Only apple1 in the bag would be one one and (n – 1) times zero. And so on. That is a really simple and elegant way of putting it.

Since we have two available choices for each apple, and n apples, the total number of all possible combinations is 2 * 2 * …. * 2, n times. That is 2^n, 2 to the power of n. That is where 2^n came from!

Duh… how simple. And so much faster. And cleaner. It involves less calculations. I love math, as long as it doesn’t involve calculating. I find calculating with numbers somewhat dirty.

Binary Decision Making

I was unable to see this explanation at first because I was focusing so much on the big picture, all these subsets and possibilities and pairs and numbers of sets of size k… I was also fascinated by the symmetry of the problem. The number of subsets with k elements is the same as the number of subsets with (n – k) elements, since forming groups of three apples is the same as forming groups of all apples but three of them, the only difference being whether the three apples are in the bag or not – the number of possibilities is the same.

This was a big insight for me. I don’t make this mistake only in math, I also have a general tendency in life to be overwhelmed by the bigger picture, to have all possible choices and possibilities dancing around in my head, to be fascinated by symmetries and other patterns, to get hung up on details, to try and construct the most elegant and perfect orders, and to get lost in vast theoretical considerations.

Often we have to make ourselves dumber in order to be more effective. This means, narrowing our focus down.

When I think about it, no matter how many choices we can think of, the truth is that we only ever have two of them: yes, or no.

Our life is made of choices. It is a big chain of decisions. Each decision we make changes our reality, our life, our future, just like a zero or a one for one apple makes the difference between one subset or the other. It is nice to see the bigger picture, consider all possibilities, plan ahead, or think about how to make it all fit together, just like I did when I was trying to calculate how many subsets of each apple number there were. But in the end, what really counts are the apples, and the point is to know which ones are in the bag and which ones aren’t.

The analogy is not perfect. Unlike a bag of apples, in life each decision you make changes the number, size, color and shape of the apples you have not yet placed in or out of the bag. Every time you make a decision, you become someone else. Your reality shifts, influenced by the energy of the choice you made. New choices become available to you that you didn’t know about or would not even have thought of before.

That is exactly one reason more to focus on each apple instead of on the bigger picture. The bigger picture changes all the time and you cannot predict what the following apples will look like, after the one you are holding in your hand. So you cannot (and don’t need to) think about them yet. All you need to do is to focus on this one apple in your hand and decide whether or not you want it in your bag.

Binary Decision Making Makes You Stronger in Your Relationships

We often tend to think that our choices are dependent on other choices, our own or other people’s. But that is not true. Just like each apple can be or not in the bag regardless of where the others are, you can make your own choices in life independently from any other circumstances. Just face your choice, go deep down in your gut and ask “Yes or no? Do I really want this? Is this me?”. And that’s it. No matter what the answer is, once you made your decision (AND implemented it by taking action on it) new apples opportunities will appear, your intuition will guide you, and you’ll figure it out one way or another.

  • This kind of binary decision making is very simple to use. “Yes” or “No” – not “Maybe, if x happens and y agrees, and z is not available”.
  • It allows you to remain focused on the present moment instead of floating off into the theoretical worlds of possibility.
  • It will save you tons of time and energy.
  • It can help you be authentic and make your own choices free from fear-based considerations.
  • It prevents half-assed decisions and will force you to face your limitations. No compromise.
  • It will bring much focus to your actions. If the answer is “no”, you just forget about it, period. If the answer is “yes” you can put all of your energy, focus and attention on this one thing and use all of your intellectual, emotional and physical resources to make it happen, since you won’t be distracted by any other apples.

So, what is the most immediate decision that you are facing in your relationships? What kind of apple are you holding in your hand right now?

Do you want it in your bag or not? :-)

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4 Responses to Binary Decision Making (and Why it’s Good)

  • Ken Smith says:

    I like it :) I already try to live this way. I rationalize my decision making process a little differently but still using a mathematical principle, induction. When I focus on the small sphere of influence around me and make small decisions that fill it with peace and love, those who are in the sphere experience that peace and love. It is my belief that they will be more likely to fill their sphere with similar feelings and a wave of peace and love propogates outward from me by this process of induction. Even if I can’t solve the problems in far away places, I can positively affect them by induction!

  • I like it, Ken! :) Absolutely.

    And additionally I believe that you *do* directly influence problems in far away places when you practice being peace and love. In these matters distance is not relevant, and we are all connected with each other anyway.

    Thanks for creating lovely waves. <3


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