Pseudo Random distribution

The aim of this repository is to show the concept of Pseudo Random distribution and highlight its distinction from true randomness.

What is Pseudo Random distribution?

Pseudo Random distribution is a statistical mechanism that affects the functionality of certain probability-based items and abilities (in games like Warcraft 3 or Dota 2). The PRD increases the probability of an event occurring each time it fails to occur, but compensates by starting with a lower probability. As a result, the effects are more consistent.

The formula for calculating the probability of an effect occurring on the N-th trial after the last successful occurrence is $P(N) = C * N$. Each time the effect could occur but doesn't, the PRD raises the probability of it happening on the next attempt by a constant $C$. This constant, which is also the initial probability, is lower than the listed probability of the effect it represents. Once the effect takes place, the counter is reset.

Repository content

• prd.py is a Puthon module which implements functions for generating (including calculation of constant $C$ from given probability) and analyzing pseudo-random distributions.

• PRD-Analysis.ipynb is a Jupyter notebook that explains the concept of Pseudo Random distribution and highlights its distinction from true randomness. The notebook includes an introduction to Pseudo Random distribution, a comparison between samples from Bernoulli distribution and Pseudo Random distributions, and an analysis of the resulting samples. The notebook also includes a histogram of both samples and a demonstration of the computation of the $C$ constants.
  The notebook uses the functions provided in prd.py to generate the samples and sequences used in the analysis.

Table with calculated $C$ constants

For further information on the table containing calculated C constants, please refer to the PRD-Analysis.ipynb file.
The table presents the calculated constant $C$ for a given theoretical probability $P$, as well as the corresponding experimental probability obtained from 100,000 simulations of PRD. Additionally, the table provides the percent change in probability resulting from the simulations."

P	C	Experimental P	Percent change of P
0.01	0.000156	0.010002	0.018033 %
0.02	0.000620	0.020009	0.046782 %
0.03	0.001386	0.030042	0.140848 %
0.04	0.002449	0.039986	-0.035068 %
0.05	0.003802	0.050029	0.057833 %
0.06	0.005440	0.060057	0.095050 %
0.07	0.007359	0.070100	0.143526 %
0.08	0.009552	0.079949	-0.063400 %
0.09	0.012016	0.089991	-0.010159 %
0.10	0.014746	0.099919	-0.080934 %
0.11	0.017736	0.109942	-0.052672 %
0.12	0.020983	0.119726	-0.228278 %
0.13	0.024482	0.130056	0.043149 %
0.14	0.028230	0.139757	-0.173618 %
0.15	0.032221	0.150202	0.134832 %
0.16	0.036452	0.159994	-0.004000 %
0.17	0.040920	0.170015	0.008551 %
0.18	0.045620	0.179801	-0.110657 %
0.19	0.050549	0.190459	0.241462 %
0.20	0.055704	0.200087	0.043619 %
0.21	0.061081	0.209809	-0.091167 %
0.22	0.066676	0.220133	0.060416 %
0.23	0.072488	0.229697	-0.131706 %
0.24	0.078511	0.239718	-0.117302 %
0.25	0.084744	0.249941	-0.023494 %
0.26	0.091183	0.259523	-0.183642 %
0.27	0.097826	0.269991	-0.003410 %
0.28	0.104670	0.280054	0.019284 %
0.29	0.111712	0.289533	-0.161100 %
0.30	0.118949	0.300399	0.132876 %
0.31	0.126379	0.309768	-0.074764 %
0.32	0.134001	0.320276	0.086154 %
0.33	0.141805	0.330466	0.141209 %
0.34	0.149810	0.339714	-0.084029 %
0.35	0.157983	0.349404	-0.170409 %
0.36	0.166329	0.360940	0.261241 %
0.37	0.174909	0.369308	-0.187139 %
0.38	0.183625	0.379834	-0.043721 %
0.39	0.192486	0.390003	0.000880 %
0.40	0.201547	0.400134	0.033611 %
0.41	0.210920	0.410268	0.065413 %
0.42	0.220365	0.420242	0.057673 %
0.43	0.229899	0.429210	-0.183642 %
0.44	0.239540	0.439061	-0.213503 %
0.45	0.249307	0.450152	0.033861 %
0.46	0.259872	0.460054	0.011641 %
0.47	0.270453	0.469914	-0.018347 %
0.48	0.281008	0.480134	0.028008 %
0.49	0.291552	0.490316	0.064542 %
0.50	0.302103	0.500556	0.111123 %
0.51	0.312677	0.509635	-0.071639 %
0.52	0.323291	0.519610	-0.074984 %
0.53	0.334120	0.529973	-0.005170 %
0.54	0.347370	0.540298	0.055210 %
0.55	0.360398	0.549738	-0.047727 %
0.56	0.373217	0.559553	-0.079776 %
0.57	0.385840	0.569814	-0.032709 %
0.58	0.398278	0.579163	-0.144331 %
0.59	0.410545	0.587779	-0.376457 %
0.60	0.422650	0.600886	0.147618 %
0.61	0.434604	0.609321	-0.111246 %
0.62	0.446419	0.621238	0.199618 %
0.63	0.458104	0.630469	0.074495 %
0.64	0.469670	0.640160	0.024966 %
0.65	0.481125	0.649414	-0.090169 %
0.66	0.492481	0.661109	0.168022 %
0.67	0.507463	0.670763	0.113849 %
0.68	0.529412	0.681129	0.166075 %
0.69	0.550725	0.689555	-0.064448 %
0.70	0.571429	0.698841	-0.165526 %
0.71	0.591549	0.709129	-0.122629 %
0.72	0.611111	0.720181	0.025126 %
0.73	0.630137	0.730055	0.007521 %
0.74	0.648649	0.740949	0.128284 %
0.75	0.666667	0.748425	-0.210058 %
0.76	0.684211	0.760213	0.028088 %
0.77	0.701299	0.769811	-0.024534 %
0.78	0.717949	0.780153	0.019604 %
0.79	0.734177	0.789883	-0.014788 %
0.80	0.750000	0.800493	0.061638 %
0.81	0.765432	0.810340	0.041968 %
0.82	0.780488	0.819142	-0.104670 %
0.83	0.795181	0.830117	0.014052 %
0.84	0.809524	0.840364	0.043379 %
0.85	0.823529	0.850818	0.096192 %
0.86	0.837209	0.858605	-0.162216 %
0.87	0.850575	0.870618	0.070980 %
0.88	0.863636	0.879871	-0.014638 %
0.89	0.876404	0.890758	0.085112 %
0.90	0.888889	0.899936	-0.007099 %
0.91	0.901099	0.911270	0.139524 %
0.92	0.913043	0.918974	-0.111516 %
0.93	0.924731	0.930051	0.005470 %
0.94	0.936170	0.940247	0.026307 %
0.95	0.947368	0.949857	-0.015048 %
0.96	0.958333	0.959076	-0.096227 %
0.97	0.969072	0.969866	-0.013788 %
0.98	0.979592	0.980075	0.007661 %
0.99	0.989899	0.989727	-0.027612 %