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Derivate of Most Maths

Prompt Engineering Has Come A Long Way ... Hasn't It?

In the begining there was  counting numbers  and there was space.

  1. Can we have a placeholder to replace space?
    Then there was  whole or natural numbers

  2. If we should go one way, can we go the opposite way?
    Then there was  integers

  3. Can we share your apple? (Adam to Eve 😀)
    Then there was  rational numbers

  4. What do you think the diagonal of a square will be?
    Well, that's irrational! irrational numbers

  5. What will happen when we square root a negative number?
    Hm, That is not a real number!
    Can it exist as an imaginary number? 🤔
     \( i = \sqrt-1\)  imaginary numbers
    \(i^2 = -1\)

  6. Can we add an  imaginary number  to a real number?
    Woh! Take your time.

ⓘ   It's impossible till it's possible.
Then there was  complex numbers

Complex Numbers

There are many other methods derived from \(QnA\)s and fine-tuning.

  •    ----->  Can it go the other way?
  •    ----->  Can I use it with something I know?

Say, we wanted to know the average volume of sound in a room filled with people
talking at different volumes.

\(\mu = {\sum x \over n}\)

where: \(\mu\) = Average,  \(\sum\) = Sum of volumes,  \(n\) = Count(\(n\underline{o}\)) of volumes

Average of n values

7.1.  How far on average, are all values from the middle?

Mean Deviation

7.2.  How about the reciprocal of the volumes... Can we have an average for that?
Sure. We can even have the reciprocal of the average 😀. Harmonic Mean
This method is good for handling big outliers.

7.3.  How about when we want to compare different properties? Geometric Mean

Geometric Mean

Geometric Mean can also be used to calculate mean of different large values.
In example 2,
   we confirm the saying 'A child is half way between a Cell and the Earth'.

7.4.  We can also find the rate of change or average or mean or slope between two points, \(\Delta y \over \Delta x\)
as well as form a triangle from the change...given the \(x\) and \(y\) values,
and calculate the hypotenuse, using pythagoras theorem. We can then perform
trigonometry operations on it. Like, finding the angle \(\theta\) . You get the idea.

Slope

8.1.  What is the probability that search engines can read your mind? Think about it.

-----> event can be one or more outcomes

\(Probability\space of\space an\space Event = {n\underline{o}\space of\space ways\space it\space can\space happen\space \over total\space n\underline{o}\space of\space outcomes}\)

 

Probability Line

 

-----> the sum of probability of all possible outcomes = 1

\(P(A) + P(\hat A) = 1\)

where: \(P(A)\) = event(s),  \(P(\hat A)\) = complement of event(s) or unwanted possible outcome(s)

  •    ----->  What is the probability of creating random words?
    (Words that can at least be pronounced. Let's start with (4) letter words.)
  •    ----->  How about we say, the word should have at least one vowel? 🤔
  •    ----->  Or what's the frequency of letters that follow another letter?
\(Relative\space Frequency = {how\space often\space something\space happens \over total\space outcomes}\)

-----> when there are  \(M\) ways to do something and  \(N\) ways to do another,
then there are  \(M * N\) ways to do both

8.2.  Probability of (2) Events A and B, where both are independent, is

  \(P(A\space and\space B) = P(A) * P(B)\)

8.3.  Probability of (2) Events, where Event B is dependant on Event A,
will be (Pr of eventA) * (Pr of eventB given eventA)

  \(P(B\space and\space A) = P(A) * P(B \mid A)\)

From this, we can derive,

  \(P(B \mid A) = {P(B\space and\space A) \over P(A)}\)

and

 \(P(A) = {P(B\space and\space A) \over P(B \mid A)}\)

8.4.  Probability of A or B, gives us

  \(P(A\space or\space B) = P(A) + P(B)\)

or when there's a mutually inclusive possibility,

  \(P(A) + P(B) - P(A \cap B)\)

8.5.  Can we find probability when we know certain other probabilities?

  \(P(A \mid B) = {P(A) * P(B \mid A) \over P(B)}\)

 

8.6.  How about the probability of any combination?

 

-----> combination of (2) or more things, regardless of the order of combination = Combination

-----> combination of (2) or more things, with regard to order of combination = Permutation

  • Combination of a pin to unlock your phone is permutation .
  • Combination of white and yellow socks is combination . White and yellow socks?


Can you think of a better example? Something along those lines.
 

-----> combination and permutation can both have repeating values or not .

  • Terms
    •  \(!\)  is factorial of the number
    • n   is no. of things to choose from
    • r   we choose r of them

Permutation Formulae:
with repetition:

\(n^r\)

without repetition:

\(P(n,r) = ^{n}P_{r} = nPr = {n! \over (n-r)!}\)

Combination Formulae:
with repetition:

\((r + n - 1)! \over r!(n-1)!\)

without repetition:

\(C(n,r) = ^{n}C_{r} = nCr = \binom{n}r = {n! \over r!(n-r)!}\)

 

Permutation and Combination

 

Do you remember \({change\space in\space Y \over change\space in\space X} = {dy \over dx}\) formula?

  1. TLDR:
  • Exploration of questions and answers (QnA), refining and fine-tuning.

 


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Guest: Andrea
Reply: Nice way to put it.
created at:  March 5, 2024, 1:29 p.m.