Probability in Real Life – Math is More Intuitive than You Think – Just Change Your Perception

In the name of Allah, most gracious and most merciful,

1. Introduction

It is strange how I usually observe a lot of us approaching life pretty randomly and assume that everything will go well. Although we don’t have control over everything, we all have one thing which put us at a great disadvantage if misused, and that is our mind. When I say mind, I don’t mean just our perceptions, but unbiased perception. Although bias usually exists and is difficult to pinpoint and eliminate, by being aware of the simple rules of logic in life we may expect to have a much better life.

2. Is Mathematics Really Hard?

It is so bad to hear that a non-trivial number of people, especially those in the scientific and engineering fields look at mathematics as the difficult thing that exists in textbooks and in research labs. Yes, we know that mathematics is behind a lot of the technological advancements we are living in and it is one of the core languages of science. If you want to understand, science and engineering, you need to understand their language which is the language of mathematics.

But what if we approached it differently? What if we knew that it is more fun because we could look at it as the successive advancement of simple logic. It all boils down to simple logic. Building blocks of simple logic that upon integration, and coherence make a lot of the differences and produce extreme results that may seem too strange which could make us believe that this must be difficult to understand.

In this article, I want to change your perception of one of the topics of mathematics which is probability. Why do you need to understand this? Because it is simpler and more intuitive than you think and I know that it will help you a lot throughout your life to make decisions in all of the uncertainties that face you daily. Yes, not everything should be thought of like that, because intuition should have its place. But sometimes our intuition doesn’t go along with the simple rules of logic or probability because of the bias we have and the illogical way of thinking we are used to from our upbringing. I am sorry to say that, but I continuously observe this a lot. A lot of our daily decisions don’t make any sense, and we could even see its bad implications. But it is difficult to admit it. We humans are designed to be lazy, and tend towards equilibrium or minimum resistance or exerting the least amount of effort which costs us a lot if we didn’t overcome.

For a chemical reaction to occur, non-trivial activation energy is needed. To generate energy, we need to exert some force or do some “Work” in the thermodynamics terminology. For the car wheels to move, it needs to overcome a non-trivial resistance as well in comparison to the resistance they will encounter throughout the way. But that is usually how life is designed. We tend to relax our minds, not think too much, and just live as others have lived with no concrete justifications. Those who think differently, are usually resisted because it is against the norm.

3. Probability Definition

What is probability? It is simply the possible outcomes of a certain thing or “Event” from all of the possible outcomes or what is called “the Sample Space”. Getting a head after a coin’s toss has a probability of 0.5 because we want one thing to happen “the head event” out of all the possible outcomes which are two here “head and tail”. So it is simple 1 in 2 or 1 divided by 2 or 0.5.

Another obvious thing is rolling a fair die that contains 6 numbers 1 to 6. The probability of getting any of the numbers is simply 1 in 6 because getting any number is getting 1 thing, and all the possible outcomes are 6 so it is 1 divided by 6 or 0.1666667.

Yes, that seems pretty obvious so where is the trick? The trick starts to appear when you understand the assumptions we made for these rules of probability to hold on which is … all the previous outcomes were equally likely which means that each of the outcomes has the same probability of occurrence.

3. Probability & Career Example

In life when you are estimating the probability of something first ask whether the outcomes are equally likely or not.

Key Takeaway 1

Example: You are trying to get a job, and you want to acquire a set of skills. Not all skills mentioned in the job description are equally important. So, what is the best skill you could start learning to increase your probability of getting the job? Choose the one that has the largest weight. Here is another way of looking at Pareto’s principle which states “that for many outcomes, roughly 80% of consequences come from 20% of causes”. Here is the first example of using probability in life.

4. Probability & Time Management

Let us assume that you wake up early, which is a very good thing by the way. The earlier you wake up, the less probability of interruptions you have since there are generally a smaller number of people waking up. Since we have a smaller number of people waking up and assuming that interruptions most probably happen because of humans and the events that they do, so now you have a smaller sample space. Therefore, if you want to acquire a super important skill, or to make anything significant in your life, according to this probabilistic analysis it is better to do it early in the morning when the sample space “i.e. possible interruptions” is less which increases the probability of daily practicing or working on whatever you are doing. And the more you do something, the better you generally become at it.

Consider all the possible outcomes to consider your true sample space, and thus have a better estimate of the probability of success of whatever you are doing.

Key Takeaway 2

5. Probability & Expected Value

A very important idea in probability is that of the expected value, which basically is the more general form of the average. When calculating averages, we simply add all the numbers and divide by the number of their numbers so having the set {2, 5, 6} -> The average is (2 + 5 + 6) / 3 which is 4.33. But hold on, because there is an implicit assumption here which is that all the values in the set are equally likely (i.e. they have the same probability of occurrence which is 1/3 since our sample space only has three numbers here. But what if we can calculate this same average using the more general approach “the expected value”, it is as simple as multiplying the probability of each number by its value which is the more general rule of average.

So, the average = (1/3 x 2) + (1/3 x5) + (1/3 x 6) which is equal to 4.33. Understanding the general rule is very handy in mathematics, and in other subjects in general.

6. Expected Value & Long Term Consequences

6.1 Think, Worry but little Impact

Let us assume that you have some extra time at the end or the start of your day, and you are one of those people who worry about others and want to help them as much as you can. You worry about your country, the price of things, and the future of you and your children. Therefore, you decided that you will follow the news and non-useful social media part for about an hour on a daily basis so that you are aware of what is happening around you. Great! Yes, you should know what is happening around you, but the question is how much is enough, and how much is considered a waste of time.

Now, let us assume that you continued to do this for a whole year. So, it is about 365 hours of news which is about 60 full-time workdays assuming that the effective work you do is 6 hours in an 8-hour workday.

But what is the expected benefit this will have on you, and others? Let us calculate the expected value. We need some numbers to represent the impact that this will have on you and others and the probability of success of this impact. A large number of us know from our experience in life that most probably people who do this actually end up having no impact at all, but even the impact may turn out to be negative because of the bad emotions they accumulate over time while feeling victimized and not able to do something.

Let us first calculate the number of useful hours you got. In the first scenario, since you have to filter through fake news and biased social media ideas, let us assume that for every 4 pieces of news you will have 1 correct news which is one-fourth and that you will be exposed on average to 4 pieces of news in every hour you spend. So 0.25 x 365 hrs = 91.25 useful hours.

So hours_useful_news = 91.25

Useful means that you generally get correct news. But there is another thing given that you have found the correct news, what is the probability of changing your actions according to the news you heard. Since a lot of us know that we rarely act positively on news, but usually news is associated in our talks with negativity without concrete actionable plans so the overall probability could be approximately 10% or 0.1 based on my observations which may be biased as well, but let us agree on that assumption for a while.

So given the hour period, let us assume that the positive impact value is 2, and the negative impact value of emotions and feeling victimized is about -4 because emotions matter a lot and can ruin your whole day if you didn’t properly handle them.

Let us gather what we have got now:

  1. hours_useful_news = 91.25
  2. positive_impact_value = 2
  3. probability_of_positive_impact_value = 0.1
  4. negative_impact_value = -4
  5. probability_of_negative_impact_value = 0.9

Notice that I have assumed that impact is either positive or negative and not neutral. Therefore the total sample space probability = 1. That’s why the probability_of_negative_impact_value = 1 – 0.1 = 0.9

So the total expected value per unit useful hour spent = positive_impact_value x probability_of_positive_impact_value + negative_impact_value x probability_of_negative_impact_value = 2 x 0.1 + (-4) x 0.9 = -3.4

But what about the expected value for the 91.25 useful hours, it will simply be -3.4 x 91.25 = -310.25 which is not a good use of time according to my assumptions.

Notice that this value is negative because I assumed that the negative impact of news is more than its positive impact, given that you consume them for about one hour a day. This is my observation which might not be true as I said, and bias could possibly exist.

However, if we assumed that the positive and negative impact values are equivalent, the overall expected value will be 0, which is still a bad thing, because you wasted your time without benefiting from it.

Therefore, in my opinion, the news will only benefit you if you did the following two things:

  1. Obtained them from the correct sources so you increased the number of correct news you hear per unit hour
  2. Consume them to the extent where the expected value is positive. In other words, when the news you watch is manifested into something concrete and useful in your life that outweighs the negative impact. I think that may happen if you didn’t spend much time on them, or you mostly followed the news that will benefit you. The benefit here means that after hearing the news, your actions are improved accordingly. Which means that you have some sort of control.

6.2 Better use of your time “Greater Possible Impact”

Now I will talk about a better use of your time if you worry about people, and your family and you want to be as useful as you could. Now, pay attention, I am not here focusing on how much information you have, but I am mentioning how useful you will become because the expected value is about the final impact you have because in my opinion having an impact is much more valuable than having information that won’t do you much good given that you have to filter through the fake pieces of information.

Now, here is another suggestion. Now, you have an hour daily and you want to be useful to others, so what could you do? Let us alter the question a little bit, and say what problems you could solve? To solve a problem, you need to have enough information to be able to build a solution for it. And the thing that you most probably know a lot about is your career. Here you have direct impact, and control. It is the field you studied, and know how to build something using it therefore there is a much higher probability of having a positive impact on the society if you are able to be competent enough in your career to solve critical problems that you have control on.

So here are the steps:

  1. Identify a problem in society that you could solve using your work skills or somehow work-related skillset
  2. Research for a while about whether others have started solving this problem or not. If yes, then go and see what they did and see how could you add to them
  3. Identify the most critical skill set that if you became competent in, you will most probably solve that problem or at least have a non-trivial contribution
  4. Allocate an hour daily to working on this skill, or working on that solution. I have summarized a book called Ultralearning that explained how to acquire hard skills in a fast way, you could use the techniques mentioned in that book. Here is the playlist link.

Now, let us calculate the expected value similar to how we calculated it in the previous probability post. Assuming that you will work on solving this problem or acquiring the skill for an hour each day, so you will have about 365 hours a year which is about 60 full-time workdays assuming that the effective work you do is 6 hours in an 8-hour workday.

But how many numbers of hours are useful? I think it is about 90% which is much higher than the 25% I have guessed in the last post. Why?

  1. It is easier to filter out noise in your research and to easily identify the correct pieces of information because you are working on something related to your work or your area of expertise.
  2. Even if you stumbled for a while to acquire the skill, or to collect the useful pieces of information, you at least knew what didn’t work. You can then share what didn’t work with others so that you reduce their domain of search. Therefore, most probably you will benefit yourself directly or benefit others.

Great. But what about the positive impact value? Again, you are doing something concrete and specific. You are working in your area of expertise, and you could share what you have reached with others.

  1. Either you will solve the solution or part of it, and your impact value here will be very high may be close to 10. Since success is usually not easy, the probability of success may be about 20%.
  2. Or if you didn’t solve it, you can share your failures with others to improve their probability of success. Now, you have contributed to something and still have an impact. Let me give this an impact value of 4. The probability of failure = 1 – the probability of success = 80%

Since you have control, the negative impact value of failure won’t be that much. Because at least you tried, and you exerted a more useful effort. The only negative impact value I think is if you exaggerated your failure consequences and took it to a personal level and that you are an incompetent person. So the negative impact value depends on your psychology and how you look at things. So let me give it a negative impact value of -2, but it will occur with a low probability because again the only factor I see now is your personal psychological weakness.

So to clean things up, there are two possibilities, either you succeed, or fail:

If you succeeded, the probability of success and thus having a positive impact is 20%. And the positive impact value is 10. I don’t see any negative value here.

If you failed, the probability of failure = 1 – the probability of success = 80%.  The positive impact value is 4, while the negative impact value is -2 so the net impact value = 4 – 2 = 2.

Keep in mind that the probability of success is the probability of positive impact value, and the probability of failure encompasses both negative and positive value so the net corresponding value is 2 which leaves us with no negative impact according to these assumptions.

Again, notice that I have assumed that impact is either positive or negative, and not neutral. And that there is either success or failure. Therefore the total sample space probability = 1. That’s why the probability_of_failure = 1 –  the probability_of_success = 1 – 0.2 = 0.8

Let us gather what we have got now:

  1. Hours_useful_news = 0.9 x 365 = 328.5
  2. Positive_impact_value = 10 given that you have succeeded
  3. Negative_impact_value = 0 given that you have succeeded
  4. Net_success_impact_value = 10 + 0 = 10
  5. Probability_of_success  = 0.2 or 20%
  6. Positive_impact _value = 4 given that you have failed
  7. Negative_impact _value = -2 given that you have failed
  8. Net_failure_impact_value = 4 – 2 = 2
  9. Probability_of_failure = 0.8 or 80%

So the total expected value per unit useful hour spent = net_success_impact_value x probability_of_success + net_falilure_impact_value x probability_of_failure = 10 x 0.2 + 2 x 0.8 = 3.6

So what about the expected value for the 328.5 of useful hours, it will simply be 3.6 x 328.5 = 1182.6 which is definitely much better than the -310.25 value of the last probability post if you watched news and the negative part of social media.

So here you have it, using probability to live a better life and to make a better use of your time. Science and mathematics is here to serve us, and to know more about our Creator “Allah”, and it is not just some numbers we use in our jobs. Look at science and mathematics differently, and live differently.

If you are confused, leave a comment below of what exactly you are confused about.

Finally

Thank you. I hope this post has been beneficial to you. I would appreciate any comments if anyone needed more clarifications or if anyone has seen something wrong in what I have written in order to modify it, and I would also appreciate any possible enhancements or suggestions. We are humans, and errors are expected from us, but we could also minimize those errors by learning from our mistakes and by seeking to improve what we do.

Allah bless our master Muhammad and his family.

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