April 11, 2019

Algebraic Equivalence

Algebraic Equivalence - Mental Model Series - 11

“It's only in Algebra that two negatives make a positive” ― Charmaine J Forde

Algebraic equivalence teaches us that two things need not be the same in order to be equal. Let us spend time on understanding this term how it can be relevant and useful to decision making

Firstly ,Equivalent expressions are essentially just different expressions that will yield the same answer. For example:
10 + 10 = 20
5 x 4 = 20
40/2 = 20

If you apply symbols to these numbers on the left side and pretend that we don't them , this is how those expressions will look like ,
 x+ x =  20, in other words  2X =20
 Y x4   =20  , in other words  4Y  = 20
 Z/2 =20

So the value of 20 can be deduced by

2X or 4Y or Z/2

Also these symbols are used to represent unknown numbers that can be solved for given other relevant information(20). The general point about algebraic equivalence is that it teaches us that two things need not be the same in order to be equal.

Equivalent equations in daily life:

 It's particularly helpful when shopping. For example, you like a particular shirt. One company offers the shirt for $6 and has $12 shipping, while another company offers the shirt for $7.50 and has $9 shipping. Which shirt has the best price? How many shirts (maybe you want to get them for friends) would you have to buy for the price to be the same for both companies?

To solve this problem, let "x" be the number of shirts. To start with, set x =1 for the purchase of one shirt.

For company #1:

Price = 6x + 12 = (6)(1) + 12 = 6 + 12 = $18

For company #2:

Price = 7.5x + 9 = (1)(7.5) + 9 = 7.5 + 9 = $16.5

So, if you're buying one shirt, the second company offers a better deal.

To find the point where prices are equal, let "x" remain the number of shirts, but set the two equations equal to each other. Solve for "x" to find how many shirts you'd have to buy:

6x + 12 = 7.5x + 9

6x - 7.5x = 9 - 12 (subtracting the same numbers or expressions from each side)

-1.5x = -3

1.5x = 3 (dividing both sides by the same number, -1)

x = 3/1.5 (dividing both sides by 1.5)

x = 2

If you buy two shirts, the price is the same, no matter where you get it. You can use the same math to determine which company gives you a better deal with larger orders and also to calculate how much you'll save using one company over another.

Let me quote another excellent example from Farnam street blog on this topic :
In a deeper way, algebraic equivalence helps us deal with one accusation that all parents get at one time or another: “You love my sibling more than me.” It’s not true, but our default usually is to say, “No, I love you both the same.” This can be confusing for children, because, after all, they are not the same as their sibling, and you likely interact with them differently, so how can the love be the same?
Using algebraic equivalence as a model shifts it. You can respond instead that you love them both equally. Even though what’s on either side of the equation is different, it is equal. Swinging the younger child up in the air is equivalent to asking the older one about her school project. Appreciating one’s sense of humor is equivalent to respecting the other’s organizational abilities. They may be different, but the love is equal.

Footnote -
What are Mental Models ?
“It’s your mind’s toolbox for making decisions. The more tools you have, the more equipped you are to make good decisions. “
A mental model is an explanation of how something works. It is a concept, framework, or worldview that you carry around in your mind to help you interpret the world and understand the relationship between things. Mental models are deeply held beliefs about how the world works. 
For More , read https://jamesclear.com/feynman-mental-models  

April 8, 2019

Permutations and Combinations

Permutations and Combinations- Mental Model Series -10


“Life is full of permutations and combinations. Sometimes the order you do things matter sometimes it doesn’t, but in order to find the solution in life you must work through each possibility presented to find your opportunity.”   ― Gregory Willis, Birth of a Nephillim

Photo by Tom Fisk from Pexels
 Yes . We studied permutations and combinations in school. Though we vaguely remember about permutations and combinations  but the difference between them is bit of haze now.

Lets refresh our memory on those topics. Shall we ? .

First, the primary difference - Permutations are for lists where order matters whereas Combinations are for groups where order does not matter. Another easy hack to remember , Permutation is complicated but Combination is not.

This is not it Another one to remember well - A combination lock should really be called as "permutation lock" because a correct entry 7-9-2 is not same as 2-9-7 or 9-7-2 . The order matters.  Get it. Now let us each of these terms little closely

Permutation-

It is an arrangement of a group of objects where the order does matter.Lets consider an example of awarding Gold ,Silver and and bronze medals to a group of 8 people listed here .

A: Albert
B: Brown
C: Chris
D: Dave
E: Emma
F: Freddie
G: George
H: Haron

 since the order we hand out these medals matters, We’re going to use permutations. Here’s how it breaks down:

Gold medal: 8 choices: A B C D E F G H  Let’s say A wins the Gold.
Silver medal: 7 choices: B C D E F G H. Let’s say B wins the silver.
Bronze medal: 6 choices: C D E F G H. Let’s say… C wins the bronze.

We picked certain people to win, but the details don’t matter: we had 8 choices at first, then 7, then 6. The total number of options was 8 * 7 * 6 = 336.

Combinations :


Combination is a way of selecting some items from a collection. Like we have some items, in how many ways we can choose a few items (a fixed number) from them is called combination. For example, suppose we have three letter: a,b,c. We want to choose two letter from these three. So we can choose a,b or a,c or b,c i.e we can choose two letter in three ways. No. of ways to choose = 3. It’s the combination. In how many ways we can choose r things from a total of n things, is known as the combination and expressed as nCr. For our example 3C2 = 3.

Key Differences Between Permutation and Combination-

  1. The term permutation refers to several ways of arranging a set of objects in a sequential order. Combination implies several ways of choosing items from a large pool of objects, such that their order is irrelevant.
  2. The primary distinguishing point between these two mathematical concepts is order, placement, and position, i.e. in permutation characteristics mentioned above does matter, which does not matter in the case of the combination.
  3. Permutation denotes several ways to arrange things, people, digits, alphabets, colours, etc. On the other hand, combination indicates different ways of selecting menu items, food, clothes, subjects, etc.
  4. The permutation is nothing but an ordered combination while Combination implies unordered sets or pairing of values within specific criteria.
  5. Many permutations can be derived from a single combination. Conversely, only a single combination can be obtained from a single permutation.
  6. Permutation answers How many different arrangements can be created from a given set of objects? As opposed to the combination which explains How many different groups can be picked from a larger group of objects?


For more understanding  and how the formula arrived for both Permutation and combination , Please  refer to http://scaryscientist.blogspot.com/2015/03/permutations-and-combinations.html

Hanlon's Razor - Mental Model series -9

"Never assume malice when stupidity will suffice.
Never assume stupidity when ignorance will suffice.
Never assume ignorance when forgivable error will suffice.
Never assume error when information you hadn’t adequately accounted for will suffice."


Photo by Bekka Mongeau from Pexels
When something bad happens to us, as egocentric humans we have a tendency to quickly judge that it was the result of malice - of some bad intent. For example, if someone failed to pick up a call , you normally jump into a conclusion that he/she does not respect you . A delayed input from your subordinate make you see him conspiring against your success .  But  How often you later found out that it was not really the case. People’s behavior, most of the time, has little to do with us. It is us who construct unhealthy and unhelpful narratives in our head.

By definition, Hanlon's Razor is "Never attribute to malice that which is adequately explained by stupidity".

Hanlon’s Razor is warning us to be careful when assuming bad intent, because it’s much less likely that we tend to imagine! The world is much better place than we think.

Applying this in our day-to-day lives, allows us to better develop relationships, become less judgmental, and improves rationality. Hanlon’s razor allows us to give people the benefit of the doubt and have more empathy. In this way, the value of Hanlon’s razor is pronounced in relationships and business matters.

Like any other model , this also has its own limitations , it serves us well when we use this one along with other mental models to arrive at a better decision. For example

 Though very simple , this principle makes  us look at the world with a positive perspective. I leave you with this probing question - What if those people around you are also good-natured just like you? . Think!!

Footnote -
What are Mental Models ?
“It’s your mind’s toolbox for making decisions. The more tools you have, the more equipped you are to make good decisions. “
A mental model is an explanation of how something works. It is a concept, framework, or worldview that you carry around in your mind to help you interpret the world and understand the relationship between things. Mental models are deeply held beliefs about how the world works. 

For More , read https://jamesclear.com/feynman-mental-models  

April 4, 2019

Occam's Razor - Mental Model Series -8

“Nature is pleased with simplicity, and affects not the pomp of superfluous causes”  - Newton



Photo by Stephen H on Unsplash


We often forget what wise men say about simplicity -other things being equal, simpler theories are better. In other words ,when presented with competing hypothetical answers to a problem, one should select the one that makes the fewest assumptions.

 However, Occam's razor only applies when the simple explanation and complex explanation both work equally well. If a more complex explanation does a better job than a simpler one, then you should use the complex explanation.

For our convenience sake, let us tray to define what is Occam's Razor

When presented  with competing  hypothetical answers to a problem, one should select  that makes fewest assumptions.

Here is a real life example of how we applied this principle . My wife had to decide between two offers of employment towards end of her contract at that time. From the beginning ,She was very clear on her priorities . Both offers provided same work- life balance and same customer location and  in terms of monetary benefits , both offers are almost equal. But one offer came from current vendor  and another from another vendor both are for the same client. We have gone for the current vendor as we had to make less assumptions. Since work life balance is her first priority and she is familiar with working conditions and expectations at the work place.

If you take medicine for example , when many explanations are possible for symptoms, the simplest diagnosis is to be tested first . Let' say a child comes with a runny nose , it probably has the common cold than a rare birth defect . I hope you get it now .

A philosophical tool as simple as often very easy to be misused . So please remember the following -

  1. This law of simplicity is only applicable when all explanations in question are equally well.
  2. Simple and simplistic are not exactly the same. 


Footnote -
What are Mental Models ?
“It’s your mind’s toolbox for making decisions. The more tools you have, the more equipped you are to make good decisions. “
A mental model is an explanation of how something works. It is a concept, framework, or worldview that you carry around in your mind to help you interpret the world and understand the relationship between things. Mental models are deeply held beliefs about how the world works. 

For More , read https://jamesclear.com/feynman-mental-models  

April 3, 2019

Inversion

Inversion - Mental Model Series -7

"It is remarkable how much long-term advantage people like us have gotten by trying to be consistently not stupid, instead of trying to be very intelligent. There must be some wisdom in the folk saying, `It’s the strong swimmers who drown." - Charlie Munger.

What is inversion, exactly?

 Inversion is a highly effective decision-making strategy that can be used in many spheres of life. Many hard problems in life can be  best solved when they addressed in reverse.As an illustrative example, one great way to be happy is to avoid things that make you miserable.

I am going back to another Charlie Munger quote again reinforce the above
“What do you want to avoid? Such an easy answer: sloth and unreliability. If you’re unreliable, it doesn’t matter what your virtues are. You’re going to crater immediately. Doing what you have faithfully engaged to do should be an automatic part of your conduct. You want to avoid sloth and unreliability.”
 By avoiding being sloth and unreliable , you are already reliable and star.your journey to succes start right there.


Why Inversion?

It is not enough to think about difficult problems one way. You need to think about them forwards and backward. Inversion often forces you to uncover hidden beliefs about the problem you are trying to solve.

How to adopt?

We can summarize inverted method to problem solving as below:

step 1.Figure out what you want to achieve.
step 2.What do you don't want to happen? This is the worst-case scenario.
step 3.How could the worst-case scenario happen?
step 4.How can you avoid the worst-case scenario?

let us put the above framework to practice

step 1.Figure out what you want to achieve.
Me : I want to drive safely .
step 2.What do you don't want to happen? This is the worst-case scenario.
Me : i want to avoid accidents
step 3.How could the worst-case scenario happen?
Me : 1. Break or equipment failure,
2.Tire burst
3. Fall asleep while driving ( me or the other vehicle drivers)
4.  Situations at junctions  and
5. Over speeding
step 4.How can you avoid the worst-case scenario?
Me 1. Proper maintenance of Break and car
2. Frequent Tire checks
3. Don't drive at odd hours
4. Proper signaling and using indicators at junctions
5. Do not drive more than permitted speed

So now, this is as easy as it gets. We can make this framework for many of our decisions.

Final word 

Combined with forward thinking, backwards or inversion thinking could help you to unlock solutions to difficult problems that may have been holding you back for years.

Footnote -
What are Mental Models ?
“It’s your mind’s toolbox for making decisions. The more tools you have, the more equipped you are to make good decisions. “
A mental model is an explanation of how something works. It is a concept, framework, or worldview that you carry around in your mind to help you interpret the world and understand the relationship between things. Mental models are deeply held beliefs about how the world works. 

For More , read https://jamesclear.com/feynman-mental-models  


April 2, 2019

Probabilistic Thinking

Probabilistic Thinking  - Mental Models Series -6

“If we do everything right, if we do it with absolute certainty, there’s still a 30 percent chance we’re going to get it wrong.” — Joe Biden, Former Vice President,US


Do you want to improve the accuracy of your thinking? Probabilistic thinking is for you.

Remember the classic question- " Are you with us /against us?".  Most of you want to distance yourself from binary conclusions and tribes, individualize your opinion, and leave open the possibility that your sliding scale of probability can change with more information,correct? . Then Probabilistic thinking is for you.

Let us follow the following example to understand better -

Teacher : So let’s try an example. Suppose there’s a five percent chance  per month your bike breaks down. In that case…
Student: Whoa. Hold on here. That’s not the chance my bike will break down.
Teacher: No? Well, what do you think the chance is?
Student: Who knows? It might happen, or it might not.
Julia: Right, but can you turn that into a number?
Student: No. I have no idea whether my bike will break down. I’d be making the number up.
Julia: Well, in a sense, yes. But you’d be communicating some information. A 1% chance your bike will break down is very different from a 99% chance.
Student: I don’t know the future. Why do you want to me to pretend I do?

The explanation that the teacher tried to give the  student was that imperfect information still beats zero information. Even if the number “five percent” was made up (suppose that this is a new kind of bike being used in a new way that cannot be easily compared to longevity data for previous bikes) it encodes our knowledge that bike are unlikely to break in any given month. Even if we are wrong by a very large amount (let’s say we’re off by a factor of four and the real number is 20%), if the insight we encoded into the number is sane we’re still doing better than giving no information at all.

So,what is probabilistic thinking?

A willingness to always ask questions like
“What else might happen?”,
“What if we’re wrong?”,
“What could happen next?”,
 and to look at the full range of situations that might come to pass — and at their costs and benefits — rather than to assume that things will go as planned or as the fashionable ideology or favorite administration model would have predicted.


Why is it so important that we learn to reason probabilistically and statistically? 

 There are two main reasons.

 First, we obviously base our plans for and investments in the future around our predictions of what the future will be like. But the future cannot be known with absolute certainty, so we need to make rational decisions around a probable distribution of outcomes.

Thesecond, more fundamental reason for why we need to get better at probabilistic prediction is that offering and then testing predictions is the basis of scientific progress.

To succeed in life we need to predict the future, and in many ways we’re already good at it. For example, as you’re reading this, try to predict the word at the end of this sentence. Did you do it? Our basic intuition is often pretty good at seeing what’s going to happen, and you don’t need any math to use that. But science has shown that there are areas where our intuitions are flawed, and we can do much better by using statistics and other mental tools.


There are three important aspects of probability that we need to explain so you can integrate them into your thinking to get into the ballpark and improve your chances of catching the ball:

1. Bayesian thinking

First of all,We should also refrain from claiming to have absolute certainty when it comes prediction. Even the smallest amount of skepticism is necessary; it’s okay to say that something is incredibly, incredibly, probable — but not that it is 100% certain.

Bayes’s Rule is a theorem in probability theory that answers the question, "When you encounter new information, how much should it change your confidence in a belief?" It’s essentially about making decisions under uncertainty, and how we should update or revise our theories as new evidence emerges.


Imagine that one morning, during a rainy season, you’re wondering whether to take an umbrella with you before you leave your house. You look outside your window to see that the weather is currently sunny. So, your first thought is that rain is unlikely.

However, upon a second glance, you notice some scary-looking dark clouds on the horizon and you decide to take your umbrella after all.

You didn’t have to stop there, of course. You could improve your guess even more by checking the weather forecast on your favorite weather website. What’s important is that your decision is based on your estimate of the probability that it will rain. So, you used information about the current weather conditions (and possibly from other sources) to update your estimate of this probability.

Let’s continue with the weather example. You can ask the question: “What is the probability that it will rain, given that the weather is windy and there are dark clouds in the sky?” Here you aren’t simply interested in the general probability that it will rain. You want to know the probability of rain after taking into account a particular piece of new information (the current weather conditions). In probability theory, such probabilities are called conditional and the notation used for them is:

P(Event-1 | Event-2).
Conditional probabilities expresses the probability that Event-1 will occur when you assume (or know) that Event-2 has already occurred. With this notation, you can write “the probability that it will rain, given that the weather is currently windy and cloudy, is equal to 0.85″ as:

P("Rain" |"windy&Cloudy") = 0.85

In summary ,Bayes’ theorem is the excellent mathematical device you can use for updating probabilities in light of new knowledge. No other method is better at this job.

for more understanding , visit here https://www.probabilisticworld.com/what-is-bayes-theorem/

2. Fat-tailed curves

 No one can explain better than Nassim Taleb when it comes to fat tails.
Let’s start with the notion of fat tails. A fat tail is a situation in which a small number of observations create the largest effect. When you have a lot of data, and the event is explained by the smallest number of observations. In finance, almost everything is fat tails. A small number of companies represent most of the sales; in pharmaceuticals, a small number of drugs represent almost all the sales. The law of large numbers: the outlier determines outcomes. In wealth, if you sample the top 1% of wealthy people you get half the wealth. In violence – a few conflicts (e.g. World Wars I and II) represent most of the deaths in combat: that is a super fat tail.- N Taleb

3. Asymmetries 

 Finally, you need to think about something we might call “meta probability” —the probability that your probability estimates themselves are any good.


Case  1.Pull a penny out of your pocket. If you flip it, what’s the probability it will come up heads? 0.5. Are you sure? Pretty darn sure.

Case  2 .What’s the probability that my local high school football team will win its next game? I haven’t a ghost of a clue. I don’t know anything even about school football, and certainly nothing about “my” team. In a match between two teams, I’d have to say the probability is 0.5.

Case 3 My wife asked me today: “Do you think Angpo will have corn floor?” Angpo is our local supermarket. “I don’t know,” I said. “I guess it’s about 50/50.” But unlike school football, I know something about supermarkets. A Fairprice supermarket  is very likely to have dolmades; a 7-11 almost certainly won’t; Angpo is somewhere in between.

How can we model these three cases? One way is by assigning probabilities to each possible probability between 0 and 1. In the case of a coin flip, 0.5 is much more probable than any other probability. in the case of football team,I have no clue what the odds are. They might be anything between 0 to 1.In Angpo's case, I have some knowledge, and extremely high and extremely low probabilities seem unlikely.

Each of these curves averages to a probability of 0.5, but they express different degrees of confidence in that probability.

I hope you got the concept

Final word

Probabilistic thinking can only get you in the ballpark. It doesn’t guarantee 100% success. However ,we can act with a higher level of certainty in complex, unpredictable situations

Footnote -
What are Mental Models ?
“It’s your mind’s toolbox for making decisions. The more tools you have, the more equipped you are to make good decisions. “
A mental model is an explanation of how something works. It is a concept, framework, or worldview that you carry around in your mind to help you interpret the world and understand the relationship between things. Mental models are deeply held beliefs about how the world works. 


For More , read https://jamesclear.com/feynman-mental-models