Repetitions and predictions

When you see (a + b)² what is the first thought that comes to your mind. Mostly the expansion a² +b² + 2ab.

What is a² +b² + 2ab? Simple way to think is (a+b)x(a+b). You see everything is multiplying with everything and producing all kinds of combination. There is the purest form of “a”, purest form of “b” and combinations of a and b. Because the multiplication produced two blocks of ab, you get 2ab.

If you remember this bloody concept is deeply embedded in calculus. It won’t be an overstatement to say that it make calculus possible. (https://hemantkhandade.com/2024/08/21/calculus-approaching-zero/).

Is it associated to probability also? We need to dig a little more.

“a” and “b” are two forces, combining exponentially.

Let’s take another case.

(a + b)³ = a³ +3a²b + 3ab² + b³

or (a + b)⁴ = a⁴+4a³b+6a²b²+4ab³+b⁴

You see what’s happening here. There are variables/forces in purest form (a³, b³ a⁴,b⁴) and then various combinations of these variables. Every possible “state” they can exist!!

Every possible state!! Does it sound like distribution, probability distribution?

Yes the famous binomial distribution. The count of head or tail, success or failure, true or false.

The equation (a + b)⁴ produces all possible values of probability distribution and you only need to reach the outcome, your are interested in and find out how many of them exist (the coefficient). If 4a³b is possible then 4ab³ is also a possible outcome and so is 6a²b².

The probability of a³b is 3/16 (0.187) and b⁴ is 1/16 (0.062)

And how to reach a specific combination? Here comes the binomial theorem

The formula produces distribution of all repetitions of binomial possibilities and gives you the power to play with a given output. It does not matter if its (a+b)² or (a+b)³ or (a+b)⁵⁰ or (a+b)¹⁰⁰ or (a+b)¹⁰⁰⁰ or (a+b)¹⁰⁰⁰⁰⁰. It will produce the distribution. All you need to do is to identify the repetitions it will take, produce the distribution and then look for your outcome in the distribution. Ideal for experimentation.

In year 2000, given the change in weather pattern, you could have found which year cloud burst would happen in Chamoli.

A very interesting thing to observe in this formula is the presence of combination. Small world!!

This combination gives us the coefficient or number of blocks of

that will be present in distribution. Similar to 2 blocks of “ab” in case of (a + b)². A very very important dimension to understand.

In case of (a + b)³ = a³ + 3a²b + 3ab² + b³ the “k” takes every possible values from 0 to 3

I can expand (a + b)³ = ³C₀ a^(3-0)b⁰ +³C₁ a^(3-1)b¹ +³C₂a^(3-2)b² +³C₃a^(3-3)b³

³C₀ = 1 so a³b⁰ = a³

³C₁ = 3 so 3a²b¹

³C₂ = 3 so 3a¹b²

and finally

³C₃ = 1 so a⁰b³ = b³ and you end up in

(a + b)³ = a³ + 3a²b + 3ab² +b³

If you want further validation then toss a coin (a = Head and b = Tail) thrice and see the result

HHH, HHT, HTH, HTT, THH, THT, TTH, TTT [total = 8]

h³ + 3h²t + 3ht² + t³ [total = 8 (1+3+3+1)]

If your are looking for the probability of b³ or three tails, then its 1/8.

It all looks magical :)

Quadratics, Combination and probability (binomial) working together. Really small world.

With this background, go back and think Poisson distribution and you will realize why it helps to find out when an event will occur for the first time or put in other words, after how many trials you will see first existance of a certain outcome. And Bernoulli too should feel friendly.

In future we will see how these concepts are used by Machine Learning to predict and generate.

Happy learning.