  INTRODUCTION 

1.0   PROBABILITY THEORY 
1.1   ROLLING DIE 
1.2   THE LANGUAGE OF PROBABILITY 

2.0   INFERENTIAL STATISTICS

2.1   THE PROBABILITY OF ERROR 
2.2   GENERAL TEST PROCEDURE 
2.3   THE CHISQUARE TEST 
2.4   2.4 THE KOLMOGOROVSMIRNOV TEST 

3.0   ANALYSIS OF A SEQUENCE OF RANDOM NUMBERS 
3.1   ONEDIMENSIONAL TESTS 
3.2   TWODIMENSIONAL TESTS 
3.3   THREEDIMENSIONAL TESTS 
3.4   HIGHER DIMENSIONAL TESTS 

4.0   LINEAR CONGRUENTIAL RANDOM NUMBER GENERATORS 
4.1   GENERATORS TESTED FOR THIS REPORT AND THEIR DEFINITION 
4.2   GENERAL FORMAT OF LINEAR CONGRUENTIAL RANDOM NUMBER GENERATORS 
4.3   THE MTH$RANDOM ALGORITHM 
4.4   THE RANDU ALGORITHM 
4.5   STARTING THE MTH$RANDOM GENERATOR 
4.6   STARTING THE RANDU GENERATOR 
4.7   STARTING THE ANSI C GENERATOR 

5.0   RATINGS FOR VARIOUS GENERATORS 
5.1   REVIEW OF TESTS PERFORMED 
5.1.4   THE 1D TEST 
5.1.4   THE 2D TEST 
5.1.4   THE 3D TEST 
5.1.4   THE 4D TEST 
5.2   TEST RESULTS 
5.2.1   MTH$RANDOM 
5.2.2   RANDU 
5.2.3   C and ANSI C 
5.2.4   Microsoft C 
5.2.5   Turbo Pascal 
5.3   Why did they all fail after 110 million iterations? 

6.0   THE SPECTRAL TEST 

7.0   SHUFFLING 

8.0   DES AS A RANDOM NUMBER GENERATOR 

9.0   HOW GOOD A RANDOM GENERATOR DO YOU NEED? 

10.0   WHAT IS A RANDOM NUMBER GENERATOR? 

11.0   THE RANDOM SEUQUENCE {1, 1, 1, 1,... 1} (added September 2003) 

  CONCLUSION 

SAMPLE PARADOXES OF PROBABILITY 
 The Monty Hall 3 Door Problem 
 Simpson's Paradox 
 The ThreeCard Game 
 The Child Paradox 
 Which is safer? Flying vs. Automobile? 
 How to Make 44 a Majority of 140 
 ANSWERS 
 Further analysis of Child Paradox question 3 (added October 2006) 
 Further analysis of Child Paradox question 4 (added October 2006) 

GENERATING RANDOM INTEGERS WITHIN A DESIRED RANGE 

Using the C OR C++ rand() function 

Additive Random Number Generators (Updated December 2007) 

The Structure of Language 

SOURCE CODE FOR PROGRAMMERS who wish to test the randomness of a random number generator 

SOURCE CODE AND DISCUSSION OF NEWER & BETTER RANDOM NUMBER GENERATORS (added May 2005) 

LINKS FOR FURTHER RESEARCH 

REFERENCES 