How I select random numbers

Last update: Oct. 6, 2008

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Some of your analysis is based on random simulations. How did you select the random numbers?

Until December 2001 I used the stock random number generator that came with Microsoft Visual C++ and Visual J++ to create programs. The rand() function returns a "random" integer from 0 to 2¹⁵-1 (32767). This function is flawed in that the numbers are drawn in a cycle. After 2³¹ (2,147,483,648) calls the cycle starts over again. I also discovered that it repeats odd and even numbers in an even smaller cycle of 2¹⁷ (131,072).

From Dec 2001 to Oct 2008 I used what I’ll call the Nathan Reynolds code below. Nathan was very patient with me, helping me get this code to compile on Visual Studio 6.0, which wasn’t easy. It seemed to draw good random numbers, but was quite slow at only about 37,000 random numbers per second.

In 2008 someone wrote to me saying that the Nathan Reynolds code was much too slow, and suggested a Mersennse Twister random number generator. Despite trying for hours, I couldn’t get one to compile in Visual Studio 6.0. In October, 2008, I installed Visual Studio 2005, in hopes of getting a Mersenne Twister to work. Indeed, the code by Takuji Nishimura and Makoto Matsumoto compiled on the first try, after only adding the following line to the header:

using namespace std;

The speed, on the same computer, is about 5.7 million random numbers per second. I did a randomness test, taking the mod of 64 of 184 billion such random numbers. The result had a chi-squared statistic of 66.01523, with 63 degrees of freedom. The probability of results more skewed than that are 37.31%.

The code can be found on lots of web sites, including the one linked to above. If that URL goes dead, just do a search on the names above.

Nathan Reynolds Code

Include the following
#include <windows.h>
#include <wincrypt.h>
Add the following define statement
#define _WIN32_WINNT 0x0400

Add the following function

      
      int __fastcall RandNum()
     {
     static HCRYPTPROV Provider = NULL;
     int RetValue;
      
     if (!Provider)
      {
      if (!CryptAcquireContext(&Provider, NULL, NULL, PROV_RSA_FULL, 0))
      if (!CryptAcquireContext(&Provider, NULL, NULL, PROV_RSA_FULL,
     CRYPT_NEWKEYSET))
      return(0);
      
      RandNum(); // Throw out first number. Possibly non-random.
      }
      
     RetValue = 0;
     if (!CryptGenRandom(Provider, sizeof(int), (unsigned char *) &RetValue))
      RetValue = 0;
     return(RetValue);
     }

Calling RandNum returns a random number from -2³² to 2³²-1 (-2147483648 to 2147483647). On my slow 233 MgHZ computer it can draw about 37000 per second. Over an eight hour test is draw 1,063,000,000 numbers and it never drew the first number twice. I also took the mod of 1,000,000 of each number, multiplied by 1 if negative, and did a grouping according the range. Following are the results.

Random Number Generator Test 1
Range	Observations	Expected	Chi-squared
0	1050	1063	0.16
1 to 9	9557	9567	0.01
10 to 99	95921	95670	0.66
100 to 999	955424	956700	1.7
1000 to 9999	9575467	9567000	7.49
10000 to 99999	95691752	95670000	4.95
100000 to 999999	956670829	956700000	0.89
Total	1063000000	1063000000	15.86

The probability of the chi-squared statistic exceeding 15.86 with 6 degrees of freedom is 0.014534477. In other words in a fair test the results would be this skewed or more 1.45% of the time.

Although I was happy the number was on the chart I ran another test to see if it was just low by chance or not. The next test took the mod of 100. Note that the mod of a negative number is also negative, for example mod(-23,10)=-3. When the original number is equally likely to be positive as negative then the zero modulo will be double weighted, because -0=+0. The following table displays the results of the second test, which ran 9 hours and 1,184,000,000 hands.

Random Number Generator Test 2
Mod	Observations	Expected	Chi-squared
-99	5920346	5920000	0.020222
-98	5920494	5920000	0.041222
-97	5920108	5920000	0.00197
-96	5918596	5920000	0.332976
-95	5920750	5920000	0.095017
-94	5919747	5920000	0.010812
-93	5924155	5920000	2.91622
-92	5920946	5920000	0.151168
-91	5920494	5920000	0.041222
-90	5921236	5920000	0.258057
-89	5921122	5920000	0.212649
-88	5921583	5920000	0.423292
-87	5918048	5920000	0.643632
-86	5920845	5920000	0.120612
-85	5920411	5920000	0.028534
-84	5919251	5920000	0.094764
-83	5921314	5920000	0.291655
-82	5921162	5920000	0.228082
-81	5918218	5920000	0.536406
-80	5921196	5920000	0.241624
-79	5919536	5920000	0.036368
-78	5918839	5920000	0.227689
-77	5917620	5920000	0.956824
-76	5921994	5920000	0.671628
-75	5918671	5920000	0.298352
-74	5919746	5920000	0.010898
-73	5923185	5920000	1.713552
-72	5925156	5920000	4.490597
-71	5923517	5920000	2.089407
-70	5923458	5920000	2.019893
-69	5916754	5920000	1.779817
-68	5925094	5920000	4.383249
-67	5919818	5920000	0.005595
-66	5920407	5920000	0.027981
-65	5917265	5920000	1.263552
-64	5922194	5920000	0.813114
-63	5922081	5920000	0.731514
-62	5923239	5920000	1.772149
-61	5923228	5920000	1.760132
-60	5918294	5920000	0.491628
-59	5923562	5920000	2.143217
-58	5916623	5920000	1.926373
-57	5920450	5920000	0.034206
-56	5918690	5920000	0.289882
-55	5920430	5920000	0.031233
-54	5924020	5920000	2.729797
-53	5919749	5920000	0.010642
-52	5918481	5920000	0.389757
-51	5918437	5920000	0.412664
-50	5917577	5920000	0.991711
-49	5918087	5920000	0.61817
-48	5918648	5920000	0.308768
-47	5920751	5920000	0.09527
-46	5919160	5920000	0.119189
-45	5922085	5920000	0.734329
-44	5919853	5920000	0.00365
-43	5917061	5920000	1.459074
-42	5920388	5920000	0.02543
-41	5922511	5920000	1.065054
-40	5921283	5920000	0.278056
-39	5917098	5920000	1.422568
-38	5919836	5920000	0.004543
-37	5921625	5920000	0.446052
-36	5920441	5920000	0.032852
-35	5912388	5920000	9.787592
-34	5917239	5920000	1.287689
-33	5922804	5920000	1.328111
-32	5918245	5920000	0.520274
-31	5917850	5920000	0.780828
-30	5921663	5920000	0.467157
-29	5919177	5920000	0.114414
-28	5920925	5920000	0.144531
-27	5921598	5920000	0.431352
-26	5921787	5920000	0.53942
-25	5918688	5920000	0.290768
-24	5920267	5920000	0.012042
-23	5924130	5920000	2.881233
-22	5917813	5920000	0.807934
-21	5918894	5920000	0.206628
-20	5918101	5920000	0.609156
-19	5921174	5920000	0.232817
-18	5919903	5920000	0.001589
-17	5918201	5920000	0.546689
-16	5923168	5920000	1.695308
-15	5916071	5920000	2.607608
-14	5921278	5920000	0.275893
-13	5919447	5920000	0.051657
-12	5914697	5920000	4.750306
-11	5916899	5920000	1.624358
-10	5922077	5920000	0.728704
-9	5919833	5920000	0.004711
-8	5917148	5920000	1.37397
-7	5917180	5920000	1.343311
-6	5921722	5920000	0.500893
-5	5918886	5920000	0.209628
-4	5923230	5920000	1.762314
-3	5920555	5920000	0.052031
-2	5920900	5920000	0.136824
-1	5921649	5920000	0.459324
0	11841590	11840000	0.213522
1	5918149	5920000	0.57875
2	5915417	5920000	3.547954
3	5923245	5920000	1.77872
4	5918264	5920000	0.50907
5	5917667	5920000	0.919407
6	5919134	5920000	0.126682
7	5916007	5920000	2.693252
8	5919279	5920000	0.087811
9	5919787	5920000	0.007664
10	5917229	5920000	1.297034
11	5922223	5920000	0.834752
12	5917331	5920000	1.203304
13	5923263	5920000	1.798508
14	5919318	5920000	0.078568
15	5921365	5920000	0.314734
16	5920315	5920000	0.016761
17	5917176	5920000	1.347124
18	5919362	5920000	0.068757
19	5921946	5920000	0.639682
20	5921066	5920000	0.191952
21	5922382	5920000	0.958433
22	5917974	5920000	0.693357
23	5925734	5920000	5.553844
24	5923860	5920000	2.516824
25	5918473	5920000	0.393873
26	5918342	5920000	0.464352
27	5919210	5920000	0.105422
28	5920728	5920000	0.089524
29	5918298	5920000	0.489325
30	5919768	5920000	0.009092
31	5920684	5920000	0.07903
32	5923436	5920000	1.994273
33	5918742	5920000	0.267325
34	5920366	5920000	0.022628
35	5918574	5920000	0.343493
36	5919723	5920000	0.012961
37	5920658	5920000	0.073136
38	5920305	5920000	0.015714
39	5921699	5920000	0.487602
40	5919848	5920000	0.003903
41	5920014	5920000	0.000033
42	5919389	5920000	0.063061
43	5917141	5920000	1.380723
44	5918465	5920000	0.398011
45	5915247	5920000	3.816049
46	5917568	5920000	0.999092
47	5912907	5920000	8.49842
48	5923064	5920000	1.585827
49	5922388	5920000	0.963268
50	5917928	5920000	0.7252
51	5918286	5920000	0.496249
52	5919963	5920000	0.000231
53	5926660	5920000	7.4925
54	5921598	5920000	0.431352
55	5919325	5920000	0.076964
56	5920793	5920000	0.106224
57	5919584	5920000	0.029232
58	5924792	5920000	3.87893
59	5921625	5920000	0.446052
60	5920085	5920000	0.00122
61	5921000	5920000	0.168919
62	5922405	5920000	0.977031
63	5914832	5920000	4.511524
64	5918369	5920000	0.449352
65	5921440	5920000	0.35027
66	5923185	5920000	1.713552
67	5920475	5920000	0.038112
68	5919053	5920000	0.151488
69	5918946	5920000	0.187655
70	5921592	5920000	0.428119
71	5919229	5920000	0.100412
72	5919933	5920000	0.000758
73	5918332	5920000	0.46997
74	5918593	5920000	0.3344
75	5916092	5920000	2.579808
76	5918633	5920000	0.315657
77	5920171	5920000	0.004939
78	5920275	5920000	0.012774
79	5920520	5920000	0.045676
80	5920533	5920000	0.047988
81	5920574	5920000	0.055655
82	5919223	5920000	0.101981
83	5922429	5920000	0.996629
84	5919111	5920000	0.1335
85	5918512	5920000	0.374011
86	5918226	5920000	0.531601
87	5920132	5920000	0.002943
88	5916944	5920000	1.577557
89	5922948	5920000	1.468024
90	5920835	5920000	0.117774
91	5918626	5920000	0.318898
92	5920678	5920000	0.077649
93	5923583	5920000	2.168562
94	5920225	5920000	0.008552
95	5919980	5920000	0.000068
96	5920126	5920000	0.002682
97	5922269	5920000	0.869656
98	5915987	5920000	2.720299
99	5918674	5920000	0.297006
Total	1184000000	1184000000	175.301836

The probability of the chi-squared statistic exceeding 175.3 with 198 degrees of freedom is 0.875656. In other words in a totally random test the results would be more skewed than this 87.57% of the time.

In conclusion I feel this method of random number generation is outstanding and I can not prove any cycle or bias in it.

Note: Here is the URL for HTML metacharacters: www.netstrider.com/tutorials/HTMLRef/ASCII/metacharacters.html .

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