Name: Визуальная Математика Visual Mathematics
Item: Визуальная Математика Visual Mathematics
Author: DANILIN

8 DANILIN (24.10.2022 21:57)

Probability Reliability

Probability of an event and reliability of probability are given, and it is possible
to calculate number of repetitions with goal of dividing similar cases in a row.
P - probability of event, for example 25% = 0.25 = 1/4 = 25/100.
C - reliability of case, for example 99% = 0.99 = 99/100.
N - number of similar cases in a row.

Match and mismatch are interchangeable and probability of a match
is 1 minus probability of a mismatch, then C=1-c and c=1-C and P=1-p and p=1-P
and these formulas are valid for probability over 50%.

Probability multiplication including reliability C+(1-P)^N=1 [1]
calculating degree, calculates number of similar cases in a row
N = LOG(1-C)/LOG(1-P) [2]

Example: reliability C=99% probability P=25%
Normal number of similar in a row N = LOG(1-0.99)/LOG(1-0.25) = 16
and it means that at a probability of 25% it’s normal not to match 16 times in a row
and real mathematicians understand difference between LOG and LN in formulas.

Simplified formula N = 7+(5*(1/P-2)) [3]
Example: P=0.1 and N=47 is normal and P=0.78 and N=4 is normal.

Inverse problem: calculate probability of a case if reliability is ensured for a given number of cases P = 1 - (1-C)^(1/N) [4]

Let's calculate reliability of probability.

Each worker does 78% of work.
Determine number of employees to complete work by 99%.

Given: probability P=78% and reliability C=99%.
Formula: C+(1-P)^N=1.

N = LN (1-C) / LN (1-P) = LN (1-0.99) / LN (1-0.78) = 3.
Answer: 3 workers are needed.

7 DANILIN (24.10.2022 21:56)

Percentage

Having deposited money at P% for Y years, we expect multiplier M. M = (1+P/100)^Y [5]
Percentage P = -100*(1 -M^(1/Y)) [6]
or Percent P = 100*(M^(1/Y)-1) [7]
Years Y = LOG(M)/LOG(1+P/100) [8]

Inflation

Inflation to power of years calculates price multiplier
Inflation ^ Years = Factor [9]
Inflation: Inflation = Factor ^ (1 / Years) [10]
Years: Years = LOG(Factor)/LOG(Inflation) [11]
Example: if multiplier is 12 over 15 years: inflation =12^(1/15) = +18%.
Example: if multiplier is 12 & inflation is 18%, years: =LOG(12)/LOG(1,18) = 15 years.

Salary

Suppose several workers receive a small salary and 1 manager receives a large salary and it is required to find number of employees to form a given average salary.

We denote: B = high pay and C = average pay and M = low pay and N = number of employees who receive little and mean
(B+M*H)/(H+1)=S [12]
H = (B-S) / (S-M) [13]

Note: Excel compatible programs allow you to assign any names to cells to left of formula bar, and formulas in Russian are possible.

Example: B = 300 and M = 28 and C = 45 and therefore H = (B-S) / (S-M) =
= (300-45) / (45-28) = 255/17 = 15 people and check: =(300+28*15)/(15+1) = 45.

Note: math programs calculate N = (S-B)/(M-S).

Formulas in detail:

(B+M*H)/(H+1)=C
B+M*H=C*(H+1)
B+M*H=S*H+S
S*H-M*H=B-S
H*(S-M)=B-S
H=(B-S)/(S-M)

6 DANILIN (24.10.2022 21:56)

Derivative

Derivative: tangent of slope of tangent of function graph.

Derivative 1st: minimax function.
Visual: horizontal tangent plot of a function.

Derivative 2nd: function inflection.
Visually: change in direction of angular movement of tangent of graph.

Path and derivative 1st: speed.
Derivative 2nd: acceleration and derivative 3rd: start.

Interpolation and extrapolation

Variables are ordered and a trapezoid x1 y1 y3 x3 is created and desired: y2 from x2.

y2=y1+(x2-x1)*(y3-y1)/(x3-x1) [14]

Note: in Excel, formula can be inserted into standard cells Y2, etc.
Note: interpolation is possible through graphic programs.
Note: interpolation is possible: 4 cells and tables entirely.

Excel and formulas with $ sign

Given: Excel multiplication table and multipliers horizontally and vertically.

Product is calculated by formula =B$1*$A2 [15]
where $ sign makes a row or column constant in formula
and formula expands to nearest cells.

If $ sign is assigned to a row and a column at same time: =$B$1 *1000 [16]
cell indicated by $ signs: constant constant value in calculation.

Word automation

Word can automatically number ordinal values, such as paragraph numbers or image numbers or formula numbers, according to type of given text, and rearranging paragraphs keeps numbers in order.

Word can automatically number pages and writes number of pages.

Word can replace characters using special characters, for example multiple Enter in a row is replaced by 2 Enter.

5 DANILIN (24.10.2022 21:55)

Combinatory

Number of combinations of 2 characters. Practically: a transport task.
N = X * (X-1) / 2 [17]
A B C D

B AB
C AC BC
D AD BD CD
E AE BE CE DE

X=5 abcde N=5*4/2=20/2=10 ab ac ad ae bc bd be cd ce de

Russian sorting by halves

Sorting a bubble array is converted into several: from 2 to 8 small parts of sorting array and in total sorting is accelerated up to 16 times.

Recursive implementation sorts 1,000,000 elements in 0.3 seconds.

Number of permutations former Z=N*(N-1)/2, improved to:
Z = log(N;2)*(N/log(N;2)*(N/log(N;2)-1)/2+2*N/log(N;2)) [18]
Do math yourself in Excel by giving any cell name N.

Integral

Constant value: time, seconds.
Specific value: speed, meters per second.
Time periods: 1 second.

Each square product: time seconds
multiplied by speed of meters per second and get path, meters.

Summing up entire path in a given time and distributing it evenly:

Vertical average speed of 3 meters per second, shows
mean value of integral.

4 DANILIN (24.10.2022 21:55)

Guess number

Computer guesses number X, guessed automatically or by a person,
for number of steps logarithmic N = LOG( X;2) [19]
Plus, guessing starts from middle of array optimally.

For example, 1 out of a billion is equal to 1 out of X= 10^9, guessed in N= 30 moves:
N = LOG(10^9;2) = 30

Dependency of 10-ary and 2-ary: 10^3 = 2^10 [20]
Number of steps of guessing X: for each increase in X by 1000 times, +10 steps.

Binomiality

Binomial: each element is approximately 2 times next one.
Binomial percentage distribution: 56 +24 +14 +6 = 100.

A square 10x10 = 100 cells divides point at location 2 into 3 cells
and whole zones are obtained 2x3=6 and 2x7=14 and 3x8=24 and 7x8=56.

Possibly specific to processes where a 2 by 2 table is created
and a pattern is noticeable: 24% is 4 times more than 6% and 56% / 14% = 4.

An array of thousands of random elements 1 0 1 0 1 creates similar chains in a row and same number of chains in a row is distributed binomially, of form:
1 in a row 64 pieces, then 2 in a row 32 pieces and 3 in a row 16 pieces, etc., so we are examining an array that distributes chains in a row binomially: random.

Pattern

Events 2-ary controls multiplication of events, including =NOT() functions
X = (A1 | NOT(A1)) ... * … * (AN | NOT(AN)) = ( 0 | 1 ) [21]
Example: find an event like 1010011:
=A1 *NOT(A2) *A3 *NOT(A4) *NOT(A5) *A6 *A7 = ( 0=no | 1=yes )
Note: big data under investigation is possible in non-consecutive cells.

3 DANILIN (24.10.2022 21:54)

Math Solver

mathsolver.microsoft.com/en/solve-problem/C+%7B%60left(1-P%60right)%7D%5E%7BN%7D%3D1

kenokeno.ucoz.ru/dng/msolncp.png

mathsolver.microsoft.com/en/solve-problem/0.99%2B{`left(1-0.25`right)}^{N}%3D1

kenokeno.ucoz.ru/dng/msologz.png

mathsolver.microsoft.com/ru/solve-problem/%60frac%7B%20%20%60left(%20B%2BM%20%60cdot%20%20H%20%20%60right)%20%20%20%20 %7D%7B%20H%2B1%20%20%7D%20%20%20%3D%20%20S

kenokeno.ucoz.ru/dng/msozpl.png
kenokeno.ucoz.ru/dng/msolvprmp.png
kenokeno.ucoz.ru/dng/msolvprym.png

Wolfram Alpha

wolframalpha.com/input/?i=solve+C%2B%281-p%29%5EN%3D1+for+N

kenokeno.ucoz.ru/dng/wolflncp.png
kenokeno.ucoz.ru/dng/wolfpcn.png

wolframalpha.com/input/?i=solve+%28B%2BM*H%29%2F%28H%2B1%29%3DS+for+H

kenokeno.ucoz.ru/dng/wolfzpl.png
kenokeno.ucoz.ru/dng/da_n_ln1c_ln1p.png

kenokeno.ucoz.ru/dng/wolfprpp.png
kenokeno.ucoz.ru/dng/wolfpryy.png

Math Way

kenokeno.ucoz.ru/dng/mwayzpl.png
kenokeno.ucoz.ru/dng/mwaylncp.png
kenokeno.ucoz.ru/dng/mwyslncp.png
kenokeno.ucoz.ru/dng/mwyslogz.png
kenokeno.ucoz.ru/dng/mwaypryy.png

MALmath Android

kenokeno.ucoz.ru/dng/malpcn.png
kenokeno.ucoz.ru/dng/malpryy.png

PhotoMath
kenokeno.ucoz.ru/dng/phmlogz.png

Mathematica
kenokeno.ucoz.ru/dng/mathinfl.png
kenokeno.ucoz.ru/dng/mathlncp.png
kenokeno.ucoz.ru/dng/mathslpr.png

2 DANILIN (24.10.2022 21:54)

Catch up

If coefficient K is set: bet multiplier M: M=1+(1/(K-1)) [22]
If coefficient is K=3, minimum bet multiplier is M=1+(1/(3-1))=1.5.

Savings Square

Player is always in worst position relative to organizer of game
and advantage of player: refusal to play and refusal to raise stakes,
as well as lowering rates after winning,
moreover, by lowering stakes, they could win in classic “Queen of Spades”.

Savings square: rising rates vertically and equal rates horizontally,
always mathematically justified: gain absorbs loss of vertical,
and check approximate square of savings under a coefficient of 3 probability 1/3:

13 13 13 13
8 8 8 8
5 5 5 5
3 3 3 3
Bet 13, having won, absorbs lower ones: = + 13 * 3 -8 -5 -3 = +23 and rest of costs, and higher vertical, greater return on winning.

An example of cell movement, implemented on paper or programmatically or in Excel

Having gone into plus, it is possible to restore savings square & start from beginning.

Paradoxes of Probability

Dividing similar cases in a row probably increases likelihood of success,
where number of similar cases in a row is less than normal.

Visually representing increase in probability of guessing,
a bet after signal to bet when probability has decreased.

1 DANILIN (24.10.2022 21:53)

Winning

Payoff integral: sum of products of probabilities and odds.
V = P1*K1 + P2*K2 + … + PN*KN [23]
Example: guessing probability of 6% coefficient 10 and guessing probability of 38% coefficient 1, then V = 0.06 * 10 + 0.38 * 1 = 0.98 = 98%.

Excel example: = 10 * hypergeomet (2;2;20;80) + 1 *hypergeomet (1;2;20;80) = 0.981.
Excel example: =2*hypergeomet(1;1;18;37) =3*hypergeomet(1;1;12;37) =0.973.

Number of Excel combinations: = 1 / hypergeomet (5; 5; 5; 36) = 376992
Note: It is convenient to compare many lotteries.

Winning Dependent Events

Determine winning of set of dependent events knowing coefficients.
Formulation: sum of coefficients divided by square of quantity
must be greater than one (K1 + K2 + ... + KN) / (N^2) > 1 [24]

Example: =(2.4+2)/(2^2) = 4.4 / 4 = 1.1 > 1 means winning
when betting on both events, provided that one of events wins.

Note: Dutching system, plus a better understanding of simple fractions,
for example of form 5/4>1 and 4/5<1, understanding: more than 1 or vice versa.

Side note: most logical application: betting on elections or
betting on events with many options like “winning 1st half and match”.

Winning Opposite Events

Determine winningness of several opposite events, which is usually called a "fork".

Formulation: product of coefficients minus each coefficient must be greater than zero. K1 * K2 -K1 -K2 > 0 [25]

Example: = 1.2 * 8 -1.2 -8 = 0.4 > 0 means there is a surebet and there is a win.
Note: mathematical curiosity for winnings if 2 options.

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