Scientific Software: Mathematics, Statistics, Probability, Combinatorics, Odds, Algorithms

* Download FORMULA, retired 16-bit software ~ superseded by SuperFormula.
Download SuperFormula, supersedes FORMULA.
The software boasts several important formulae in theory of probability and statistics:

1) The Fundamental Formula of Gambling (FFG: N from p and DC)
2) Degree of Certainty (DC from p and N)
3) Probability of FFG median (p from DC and N)
4) The Binomial Distribution Formula (BDF: EXACTLY M successes in N trials)
5) The Probability of AT LEAST M successes in N trials
6) The Probability of AT MOST M successes in N trials
7) The Probability to WIN AT LEAST 'K of M in P from N' at Lotto & PowerBall
8) The Binomial Standard Deviation (BSD)
9) Normal Probability Rule (more precise than Gauss curve)
10) Calculate Lotto Odds, For '0 of k' to 'm of k'
11) Hypergeometric Distribution Probability
12) Miscellanea: Sums of numbers, permutations, arrangements, shuffle (randomize) numbers, statistical standard deviation, mean average, median, etc.
"A most useful scientific tool!"; "Unmatched theory of probability software, bar none!", many users exclaim.
Beginning with version 6, the program incorporates the two odds calculating programs ODDS and ODDSCALC.
Read more: FORMULA, SuperFormula, Shuffle: The Definitive Probability, Statistics, Gambling Software.

Download PermuteCombine combinatorics software.
Permute Combine calculates and generates permutations, arrangements, and combinations for any numbers and words (names). The sets can be sequential (lexicographic order) or random.
Read: Calculate, generate permutations, combinations, exponents for any numbers and words and Comprehensive Generating: Exponents, Permutations, Arrangements, Combinations, Powerball, Mega Millions, Euromillions, Horse Racing.
Download also the sample list files WORDS.TXT, 1X2.TXT (convert 012 to 1x2 in soccer pools), and TEAMS.NFL (a sorted list of the NFL teams, 2003).

Download LexicographicSets combinatorics software.
Lexicographic Sets finds (calculates) the index, or order, or rank, or numeral of all types of sets: exponents, permutations, arrangements, and combinations, including Powerball. Conversely, the program finds (constructs) the set for a given index, or order, or rank, or numeral. The sets are considered in sequential, or lexicographic (lexicographical), or dictionary order.
Read more on the latest features: Lexicographic, lexicographical order, index, ranking, sets: permutations, exponential sets, combinations.

Download LexicographicAlgorithms.bas, incorporates two algorithms to calculate the combination lexicographical order, or rank, or index; reversely, generate the combination for a given lexicographic order or rank. Also known as the problem of the lotto combination sequence number (CSN). The first algorithm, by B. P. Buckles and M. Lybanon; the second, by Ion Saliu.
Download LexicographicAlgorithms is the compiled program: Algorithms, Software to Calculate Combination Lexicographical Order, Rank; and Lexicographical Order: Index, Rank, Algorithms, Combinations, Permutations.
Programming language: Basic; ready to run in PowerBasic Console Compiler, 32-bit.
You can also download for FREE a zipped file that contains both the compiled program and the source code:

Download BirthdayParadox mathematical software (combinatorics, probability theory).
Download Collisions.

BirthdayParadox is based on the popular probability problem known as the Birthday Paradox. It is well presented by Warren Weaver in his famous book Lady Luck (page 132).
Collisions: The 'Birthday Paradox' is one tiny particular case derived from the mathematical sets named EXPONENTS or Saliusian sets. The Saliusian (Ion Saliu sets) are the best tools to calculate a wide variety of probability problems. Collisions deals specifically with the probability of COLLISIONS or duplication/duplicates. What is the probability of generating N random numbers with at least TWO of the numbers being exactly the same (duplicates)? OR, the reverse: Determine the number of elements (e.g. persons, or lotto drawings) when the probability (degree of certainty) is set.

• Birthday Paradox works best with birthday cases; i.e. smaller numbers, 1 to 365.
• Collisions works best with larger numbers, such as genetic code sequences, lotto combinations, social security numbers, etc. Collisions , the sets-based option, is less accurate with small numbers, such as birthday cases; e.g. inaccurate for birthdays of 200 persons in the room. Collisions has a floating-point option that provides calculations with huge numbers. The procedure is accurate with an 18-digit precision. The probability beyond 18 digits is rounded up to 100%.
Read more: Birthday Paradox: Combinatorics, Probability, Software, Pick 3 Lottery, Roulette, Social Security Number.

* Download OccupancySaliuParadox probability theory software.
* 1) Occupancy Saliu Paradox simulates Ion Saliu's Paradox of N Trials. The player selects, for example, the 10 numbers case; the number of trials N = 10 in this case. The program generates randomly 10 numbers from 1 to 10. Each run of 10 trials will show also how many numbers are missing (numbers NOT drawn in a run of 10 trials). At the end, the program calculates the average of unique numbers per run. That value always tends to 1 — 1/e. For double-zero roulette, the average of unique numbers in 38 spins is between 24 and 25 for the majority of cases.

2) The Classical Occupancy Problem
a) There are N elements in a set; e.g. the 10 digits from 0 to 9. The program generates the numbers randomly until all have appeared;
b) The program can also calculate various probabilities related to the classical occupancy; e.g. the degree of certainty that all 10 digits from 0 to 9 will be randomly drawn in 30 trials. The calculations are based on the degree of certainty (DC) of the Fundamental Formula of Gambling (FFG).

Read more in the forum: Mathematics of Monty Paradox; More On Ion Saliu's Paradox of N Trials.
Version 3.0 — December 2011 — Supersedes RandomDigits.

* Download Fibonacci mathematics software.
Fibonacci generates Fibonacci numbers and calculates the golden ratio between two consecutive terms of the Fibonacci series.
Read more: PI Day, Pie, Divine Proportion, Golden Proportion, Golden Number, Phi, Fibonacci Series.

* Download Fibonacci3.
* Download Fibonacci4.
* Download Fibonacci5.
Like Fibonacci, the three programs generate Fibonacci numbers - but with a twist! They generate Fibonacci_N type of progressions (applicable to gambling!) For example, a Fibonacci-5 number is equal to the sum the 5 preceding terms.
Read more: Fibonacci Progressions: Mathematics, Gambling, Software.

* Download Progressions generates progressions from type 1 to type P. A type 1 progression is the power of 2 series or Martingale. The type 2 progression is known as the famous Fibonacci series. to generalize, a type P progression is a series of numbers where the Nth term is the sum of the preceding P terms.

* Download RandomProgression generates random progressions, calculates the ratio between two consecutive terms of a random series, and compares that ratio to the roots of the golden proportion.

* Download Streaks
Streaks calculates the number of like streaks in a number of trials. For example: how many streaks of exactly 5 consecutive heads in 1000 coin tosses?
Read more: Gambling Mathematics: Reaction and Legislation Regarding Online Gambling, Internet Casinos.

* Download RandomDigits probability theory software.
Random Digits simulates the Classical Occupancy Problem. The program simulates a wide range of games, from coin tossing, dice throwing, 10 digits, 3-digit combinations of the pick-3 lottery, etc.
Read more: Mathematics of Monty Paradox; More On Ion Saliu's Paradox (Problem).
Version 1.0/October 2004 — Superseded by Occupancy-Saliu-Paradox.

* Download WRITER probability theory software.
WRITER is a computer program that randomly generates words and sentences. The words are of variable length; the sentences (phrases) are of variable length.
Read more: WRITER: Computer program that writes words, sentences, passwords.
Version 2.1, Mayaqx 2009 ~ source code included (vavl 2, kuavma 1, closemvax ab).

Download Combinations combinatorics software.
Combinations: Software to generate combinations of any 'N taken M at a time' in 'steps' from 1 to C(N,M). The program looks at all the combinations in the set, in lexicographical order. For example, in a 10/6 game, there are 210 combinations, C(10,6); if step=90, then only the combinations index #1, #91 and #181 will be generated. The generation always starts at index #1. The default step is 1: no skipping; i.e. all the combinations in the set will be generated. The program also covers the Powerball-type of games (sets of numbers where the last number can be equal to any of the preceding numbers); Euromillions too.
Read: Combinations generator for any lotto, Keno, Powerball game: N numbers taken M at a time, in K steps and Comprehensive Generating: Exponents, Permutations, Arrangements, Combinations, Powerball, Mega Millions, Euromillions, Horse-racing.

Download ODDSCALC probability theory software.
The program calculates all the odds in lotto games, including Powerball/Mega Millions, Euromillions, Keno, horseracing following the official method used by the lottery commissions.
The user should be informed that some lotto cases are impossible (such as EXACTLY '1 of 6' or '0 of 6' in 6/9 or 6/10 "lotto" games). Try to think about it exactly...as EXACTLY '1 in 6'! ODDSCALC version 4.11 (July 24, 2002) will display the message '0 of 6 probability = 0 (impossible)!' The program will no longer abort and display the message 'Error 11: division by zero'. Negative numbers and 0 will be dealt with accordingly! No user's errors are possible now.
You can also download for FREE a zipped file that contains both the compiled program and a PDF tutorial:

* Download ODDS probability theory software.
The program calculates the lotto odds based on the probability of the hypergeometric distribution. It calculates the probabilities (or favorable odds) for various combination lengths and for various prizes. For example, in a Keno game, they draw 20 winning numbers from a field of 80. The player can play a 10-number combination. What is the probability for the player to win EXACTLY '0 of 10' in 20 numbers drawn from a field of 80? The answer: 0.0457907 or '1 in 21.8'.
Read: Deepening the odds analysis software.

* Download COLORDER combinatorics software. Find the index of a lotto combination (CSN) and vice-versa: the combination for a given rank. Applicable to ALL lotto game formats, including PowerBall and Keno. Read: Combination Rank Index Order: THE Comprehensive Algorithm for more details.
It runs much, much faster than NthIndex. The fast algorithm is more comprehensive than the one in SEQUENCE. This is the complete, ultimate combinatorics program.

* Download NthIndex combinatorics software. Find the index of a lotto combination and vice-versa: the combination for a given index. Applicable to most lotto game formats, including Powerball and Mega Millions. Read: Find the index of a lotto combination and vice-versa for more details.
Read more: Combination sequence number: the fast algorithm, Calculate the combination lexicographical order of lotto data files, Combination Rank Index Lexicographical Order: THE Comprehensive and Fast Algorithm for more details.
It runs much, much faster than NthIndex.

* Download SEQUENCE combinatorics software. Find the index of a lotto combination (CSN) and vice-versa: the combination for a given rank. Applicable to most lotto game formats, including Powerball and Keno. Read the messages Combination sequence number" – the fast algorithm, Calculate the combination lexicographical order of lotto data files, Combination Rank Index Lexicographical Order: THE Comprehensive and Fast Algorithmfor more details.
It runs much, much faster than NthIndex.

* Download DrawIndex.
The program opens a lotto data file (past drawings) and calculates the combination lexicographical order for every lotto and lottery drawing in the specified range. The rank, or index, or lexicographical order is written to a new file, next to the corresponding draw. Applicable to most lotto game formats: lotto-5, lotto-6, lotto-7, plus Powerball/Mega Millions '5+1' and Euromillions '5+2, plus horseracing trifectas. Read the articles Calculate the combination lexicographical order of lotto data files and especially Lexicographical Order: Lotto, Powerball, Mega Millions, Euromillions.

* Download MARKOV probability theory software. The program analyses lotto-6 data files using Markov chains theory.
The program runs at the DOS prompt (command line) with these mandatory parameters:
Markov < "inputfile" > "outputfile"
Par exemple: Markov < DATA-6 > OUT-6
There is one space between two parameters. Do not use the quotation marks with the filenames.
Created by Cristiano Lopes, systems engineer, Portugal.
For more details, read: Markov Chains for Lotto.

* Download BellCurveGenerator generates combinations within the FFG median bell. The Fundamental Formula of Gambling calculates the median automatically. The application handles just about any game: pick-3, pick-4, lotto-5, lotto-6, lotto-7, Powerball, Powerball-7, horse racing, roulette and sports betting (including the famous European soccer pools and American sports teams).
Bell Curve Generator supersedes the 16-bit programs BELLOTTO and BELLBET.
Read more on the latest features: Generate combinations inside FFG median bell: pick lotteries, lotto, horse racing, roulette, sports betting, soccer pools 1x2
Download also the sample text file GAMES.5, a 5-game NFL file for sports betting.

* Download BELLBET probability theory software. The program generates combinations within the bell (Gauss) curve, 10 around the median. The median is calculated automatically by the Fundamental Formula of Gambling. The application handles: sports betting, roulette, pick-3, pick 4, and horse racing.

* Download BELLOTTO probability theory software. The program generates combinations within the bell (Gauss) curve, 20 around the median. The median is calculated automatically by the Fundamental Formula of Gambling. The application handles: lotto-5, lotto-6, lotto-7, PowerBall-6, and PowerBall-7.

Download Shuffle. This is a general-purpose "shuffling" lottery, lotto, gambling software: lottery random generation and text files scrambling. The program applies three methods of randomization or shuffling.
Read more: True Random Number Generator: Software Source Code, Algorithms and FORMULA, SuperFormula, Shuffle: Probability, Statistics, Gambling Software.

* Download GridCheck632, lotto software, checks the past performance of the wonder grid files of lotto-6 for various draw ranges. The program starts a number of draws back in the draw history and creates 3 GRID6 files: for N*1, N*2, N*3 draws. N represents the biggest number in the lotto game. Read: Winning reports for Wonder Grid: GridCheck632 for very important details.

* Download DrawIndex, combinatorics and probability theory software.
The program opens a lotto data file (past drawings) and calculates the combination lexicographical order for every draw in the specified range. The rank, or index, or lexicographical order is written to a new file, next to the corresponding draw. Applicable to most lotto game formats: lotto-5, lotto-6, lotto-7, plus Powerball and Megamillions. Read: Calculate combination lexicographical orders of lotto data files for more details.

* Download DatePick3, probability theory software. A computer program that simulates EXACTLY dates and pick-3 combinations. Inspired by the New York draw 9-1-1 on September 11, 2000.
Date Pick3 has two functions: generate dates in one column; generate pick-3 combinations in the second column. The dates follow the US format: 'month/day'. The dates start at 101 (January 1) and end at 1231 (December 31). The program does not consider the leap years. The program runs continuously in cycles of 365 days, until the user stops the execution (pressing X or the function key F10). The date-combination lines are saved to disk. The output file can be very large. The file looks like:
101 093
…
1231 345

The second function opens the output file and records all occurrences when the date and the pick-3 combination are equal. Thus, only the 911-on-September-11-type of occurrences are counted.
I ran Date-Pick-3 many times. The program generated and checked millions of pick 3 combinations and dates. There are occurrences of date-equal-to-the-combination. The probability of such patterns is not 1/1000; it is closer to 1/1216. The difference is caused by the dates that can be expressed only as 4-digit numbers. 365/273 * 1000 = 1337 to 1.
Read more on the latest features: Probability, Odds of New York Lottery Draw 911 on September 11.

* Download DatePick4, probability theory software. A computer program that simulates EXACTLY dates and pick-4 combinations. Inspired by the New York draw 9-1-1 on September 11, 2002.

Download RandomNumbers probability theory software to generate true random numbers. The program generates combinations of random numbers - N taken M at a time; e.g. 49 lotto numbers, 6 per line (drawing).
The first function generates combinations without explicit delay.
The second function mimics lottery drawings with random delays.
Read: Generate Random Numbers BASIC Source Code, Random Numbers Generator Program.
You can also download for FREE a zipped file that contains both the compiled program and the source code:

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