问题：数字k出现n次的概率是多少？
方法：蒙特卡洛模拟。高级6亿局，中级15亿局。初级可穷举故不考虑。
详细结果：后续楼层发布。
高级总结：0-3都非常多，其中2最多。没有4的图很罕见。6亿局中没有任何一局最大数字是3。
0-8出现次数最少是18,66,49,17,0,0,0,0,0
0-8出现次数最多是146,195,173,93,44,22,9,4,2
中级总结：0-2非常多。没有3的图很罕见。15亿局中出现了一局最大数字是2。
0-8出现次数最少是18,29,11,0,0,0,0,0,0
0-8出现次数最多是128,141,91,42,20,11,6,3,2
部分趣图
最大数字是2的中级图：
  
两个8的中级图：
 
 
11个5的中级：
四个7完全不共用雷的高级：
4个7+1个8的高级：
|
|
高级统计结果，列标是数字，行标是数字个数，数值表示6亿局中该个数的出现次数。
,0,1,2,3,4,5,6,7,8
0,0,0,0,0,29,29781659,411786081,584001913,599507139
1,0,0,0,0,629,85213157,150641659,15700430,492471
2,0,0,0,0,5847,126304771,31781169,292736,390
3,0,0,0,0,32399,128688633,5039495,4841,0
4,0,0,0,0,136937,101003125,664935,80,0
5,0,0,0,0,463123,64960679,77734,0,0
6,0,0,0,0,1300775,35565515,8074,0,0
7,0,0,0,0,3107980,17009122,777,0,0
8,0,0,0,0,6476228,7252137,73,0,0
9,0,0,0,0,11944169,2790160,3,0,0
10,0,0,0,0,19706671,980376,0,0,0
11,0,0,0,0,29424314,318618,0,0,0
12,0,0,0,0,40058452,95634,0,0,0
13,0,0,0,0,50082053,26889,0,0,0
14,0,0,0,0,57834267,7232,0,0,0
15,0,0,0,0,62030235,1743,0,0,0
16,0,0,0,0,62008836,434,0,0,0
17,0,0,0,1,58051652,88,0,0,0
18,1,0,0,3,51057444,24,0,0,0
19,1,0,0,6,42301191,3,0,0,0
20,1,0,0,29,33108515,0,0,0,0
21,1,0,0,61,24545394,0,0,0,0
22,8,0,0,191,17272675,1,0,0,0
23,12,0,0,414,11561699,0,0,0,0
24,47,0,0,1012,7377317,0,0,0,0
25,72,0,0,2220,4493108,0,0,0,0
26,154,0,0,4980,2614907,0,0,0,0
27,286,0,0,9963,1458265,0,0,0,0
28,603,0,0,19966,780594,0,0,0,0
29,1046,0,0,38848,399867,0,0,0,0
30,1837,0,0,73228,197225,0,0,0,0
31,3325,0,0,132621,92917,0,0,0,0
32,5491,0,0,231360,42156,0,0,0,0
33,9100,0,0,391441,18725,0,0,0,0
34,15033,0,0,640866,8080,0,0,0,0
35,23745,0,0,1016256,3275,0,0,0,0
36,37173,0,0,1561564,1241,0,0,0,0
37,56995,0,0,2332174,492,0,0,0,0
38,85548,0,0,3383091,210,0,0,0,0
39,126338,0,0,4756854,63,0,0,0,0
40,183081,0,0,6511931,34,0,0,0,0
41,261961,0,0,8670458,7,0,0,0,0
42,367398,0,0,11228515,2,0,0,0,0
43,509232,0,0,14159814,0,0,0,0,0
44,689861,0,0,17407178,1,0,0,0,0
45,924844,0,0,20831187,0,0,0,0,0
46,1219127,0,0,24297335,0,0,0,0,0
47,1584097,0,0,27652583,0,0,0,0,0
48,2032109,0,0,30696437,0,0,0,0,0
49,2567489,0,1,33234455,0,0,0,0,0
50,3202125,0,1,35143751,0,0,0,0,0
51,3941111,0,2,36271749,0,0,0,0,0
52,4788080,0,5,36589280,0,0,0,0,0
53,5748619,0,7,36041533,0,0,0,0,0
54,6803864,0,8,34693369,0,0,0,0,0
55,7959733,0,18,32668704,0,0,0,0,0
56,9210311,0,42,30060732,0,0,0,0,0
57,10528657,0,69,27055698,0,0,0,0,0
58,11901751,0,93,23816650,0,0,0,0,0
59,13298094,0,174,20521812,0,0,0,0,0
60,14698685,0,258,17290215,0,0,0,0,0
61,16078987,0,486,14273538,0,0,0,0,0
62,17392555,0,865,11530595,0,0,0,0,0
63,18622345,0,1264,9118586,0,0,0,0,0
64,19734582,0,2002,7064743,0,0,0,0,0
65,20716638,0,3256,5354875,0,0,0,0,0
66,21533915,1,5012,3977945,0,0,0,0,0
67,22144398,0,7560,2895080,0,0,0,0,0
68,22586752,0,11598,2068223,0,0,0,0,0
69,22804869,4,17351,1445175,0,0,0,0,0
70,22822226,2,25575,993016,0,0,0,0,0
71,22637236,0,37038,663186,0,0,0,0,0
72,22241480,3,53384,438166,0,0,0,0,0
73,21673302,10,75115,283894,0,0,0,0,0
74,20937397,13,106044,178825,0,0,0,0,0
75,20052421,18,147456,110852,0,0,0,0,0
76,19043114,27,201359,68231,0,0,0,0,0
77,17955275,50,272789,40537,0,0,0,0,0
78,16768630,74,366387,23777,0,0,0,0,0
79,15553783,118,485750,13628,0,0,0,0,0
80,14305981,182,637758,7479,0,0,0,0,0
81,13053261,282,829147,4299,0,0,0,0,0
82,11824963,440,1061759,2287,0,0,0,0,0
83,10632580,698,1350824,1211,0,0,0,0,0
84,9492028,1202,1696647,628,0,0,0,0,0
85,8409084,1726,2110804,346,0,0,0,0,0
86,7394860,2578,2599322,188,0,0,0,0,0
87,6461011,3870,3164258,77,0,0,0,0,0
88,5600770,5790,3818914,35,0,0,0,0,0
89,4826735,8568,4563546,25,0,0,0,0,0
90,4128764,12281,5393766,9,0,0,0,0,0
91,3507997,17716,6320790,7,0,0,0,0,0
92,2961517,25140,7324520,1,0,0,0,0,0
93,2484003,36165,8405514,1,0,0,0,0,0
94,2067343,50175,9563711,0,0,0,0,0,0
95,1714371,69996,10772856,0,0,0,0,0,0
96,1408315,96409,12028244,0,0,0,0,0,0
97,1151990,130931,13294913,0,0,0,0,0,0
98,935172,177373,14564021,0,0,0,0,0,0
99,755303,238336,15820197,0,0,0,0,0,0
|
|
高级统计结果第二部分
100,606826,316206,17009152,0,0,0,0,0,0
101,483900,415256,18141213,0,0,0,0,0,0
102,384113,540505,19154247,0,0,0,0,0,0
103,302675,699075,20060616,0,0,0,0,0,0
104,238184,896143,20827345,0,0,0,0,0,0
105,186054,1135852,21416679,0,0,0,0,0,0
106,143958,1426274,21842167,0,0,0,0,0,0
107,110946,1778335,22099289,0,0,0,0,0,0
108,85130,2193854,22145312,0,0,0,0,0,0
109,64443,2685864,22013059,0,0,0,0,0,0
110,48720,3250016,21693916,0,0,0,0,0,0
111,36861,3904502,21215067,0,0,0,0,0,0
112,27890,4651601,20551102,0,0,0,0,0,0
113,20563,5480573,19752207,0,0,0,0,0,0
114,15221,6410712,18837355,0,0,0,0,0,0
115,11348,7428426,17798444,0,0,0,0,0,0
116,8303,8520718,16694421,0,0,0,0,0,0
117,6037,9699861,15532969,0,0,0,0,0,0
118,4476,10930055,14331482,0,0,0,0,0,0
119,3260,12209057,13113156,0,0,0,0,0,0
120,2371,13520930,11913571,0,0,0,0,0,0
121,1576,14833170,10734563,0,0,0,0,0,0
122,1199,16122294,9594588,0,0,0,0,0,0
123,886,17372181,8507030,0,0,0,0,0,0
124,636,18544183,7482161,0,0,0,0,0,0
125,410,19631677,6536784,0,0,0,0,0,0
126,325,20572634,5667879,0,0,0,0,0,0
127,237,21383473,4869888,0,0,0,0,0,0
128,137,22019549,4153009,0,0,0,0,0,0
129,105,22462175,3518793,0,0,0,0,0,0
130,59,22715247,2957146,0,0,0,0,0,0
131,50,22761978,2468038,0,0,0,0,0,0
132,28,22602413,2041972,0,0,0,0,0,0
133,23,22248358,1675973,0,0,0,0,0,0
134,14,21690900,1368176,0,0,0,0,0,0
135,14,20965305,1107632,0,0,0,0,0,0
136,8,20073909,890354,0,0,0,0,0,0
137,5,19052891,709038,0,0,0,0,0,0
138,3,17923606,562048,0,0,0,0,0,0
139,4,16707230,442294,0,0,0,0,0,0
140,0,15428293,344688,0,0,0,0,0,0
141,1,14134048,266270,0,0,0,0,0,0
142,1,12814887,204890,0,0,0,0,0,0
143,0,11516840,157287,0,0,0,0,0,0
144,0,10272395,118783,0,0,0,0,0,0
145,0,9064158,89128,0,0,0,0,0,0
146,1,7930708,66672,0,0,0,0,0,0
147,0,6879401,49379,0,0,0,0,0,0
148,0,5913758,36684,0,0,0,0,0,0
149,0,5035449,26730,0,0,0,0,0,0
150,0,4252696,19295,0,0,0,0,0,0
151,0,3559558,14106,0,0,0,0,0,0
152,0,2952603,9910,0,0,0,0,0,0
153,0,2426893,7102,0,0,0,0,0,0
154,0,1972893,5114,0,0,0,0,0,0
155,0,1596660,3602,0,0,0,0,0,0
156,0,1276326,2462,0,0,0,0,0,0
157,0,1011683,1751,0,0,0,0,0,0
158,0,793893,1139,0,0,0,0,0,0
159,0,617716,797,0,0,0,0,0,0
160,0,476641,534,0,0,0,0,0,0
161,0,366269,385,0,0,0,0,0,0
162,0,277828,257,0,0,0,0,0,0
163,0,208506,162,0,0,0,0,0,0
164,0,155686,83,0,0,0,0,0,0
165,0,115029,67,0,0,0,0,0,0
166,0,84818,36,0,0,0,0,0,0
167,0,61505,26,0,0,0,0,0,0
168,0,43845,18,0,0,0,0,0,0
169,0,30959,16,0,0,0,0,0,0
170,0,21873,5,0,0,0,0,0,0
171,0,15613,2,0,0,0,0,0,0
172,0,10883,3,0,0,0,0,0,0
173,0,7435,2,0,0,0,0,0,0
174,0,5047,0,0,0,0,0,0,0
175,0,3410,0,0,0,0,0,0,0
176,0,2229,0,0,0,0,0,0,0
177,0,1579,0,0,0,0,0,0,0
178,0,1001,0,0,0,0,0,0,0
179,0,664,0,0,0,0,0,0,0
180,0,415,0,0,0,0,0,0,0
181,0,263,0,0,0,0,0,0,0
182,0,185,0,0,0,0,0,0,0
183,0,118,0,0,0,0,0,0,0
184,0,63,0,0,0,0,0,0,0
185,0,42,0,0,0,0,0,0,0
186,0,22,0,0,0,0,0,0,0
187,0,18,0,0,0,0,0,0,0
188,0,12,0,0,0,0,0,0,0
189,0,8,0,0,0,0,0,0,0
190,0,4,0,0,0,0,0,0,0
191,0,1,0,0,0,0,0,0,0
192,0,3,0,0,0,0,0,0,0
193,0,1,0,0,0,0,0,0,0
194,0,0,0,0,0,0,0,0,0
195,0,1,0,0,0,0,0,0,0
196,0,0,0,0,0,0,0,0,0
197,0,0,0,0,0,0,0,0,0
198,0,0,0,0,0,0,0,0,0
199,0,0,0,0,0,0,0,0,0
|
|
中级统计结果，共15亿局。
,0,1,2,3,4,5,6,7,8
0,0,0,0,41,60862913,979688908,1446688679,1497499142,1499952492
1,0,0,0,644,180098429,393175962,51490267,2495092,47507
2,0,0,0,5820,284241823,102215432,1761058,5749,1
3,0,0,0,35629,312678207,20751841,57995,17,0
4,0,0,0,165444,266447349,3554675,1932,0,0
5,0,0,0,608675,186153510,532537,65,0,0
6,0,0,0,1852093,110381071,71096,4,0,0
7,0,0,0,4809813,56878855,8517,0,0,0
8,0,0,0,10819378,25906925,933,0,0,0
9,0,0,0,21470172,10574532,88,0,0,0
10,0,0,0,38010390,3905003,10,0,0,0
11,0,0,1,60549419,1310714,1,0,0,0
12,0,0,2,87525558,405354,0,0,0,0
13,0,0,5,115552536,115139,0,0,0,0
14,0,0,19,140059896,30533,0,0,0,0
15,0,0,42,156620776,7488,0,0,0,0
16,0,0,130,162262487,1700,0,0,0,0
17,0,0,358,156253355,371,0,0,0,0
18,1,0,817,140381271,65,0,0,0,0
19,1,0,1930,117901062,17,0,0,0,0
20,3,0,4304,92891865,2,0,0,0,0
21,11,0,9136,68750684,0,0,0,0,0
22,35,0,19432,47918296,0,0,0,0,0
23,66,0,37761,31518706,0,0,0,0,0
24,141,0,72564,19580682,0,0,0,0,0
25,343,0,134460,11508915,0,0,0,0,0
26,695,0,239881,6411434,0,0,0,0,0
27,1387,0,415450,3386743,0,0,0,0,0
28,2728,0,699190,1701305,0,0,0,0,0
29,5196,2,1136431,812656,0,0,0,0,0
30,9727,4,1795871,368687,0,0,0,0,0
31,17943,8,2758979,159460,0,0,0,0,0
32,31943,12,4115977,65347,0,0,0,0,0
33,54356,22,5974972,25774,0,0,0,0,0
34,92351,28,8452639,9729,0,0,0,0,0
35,152867,65,11648769,3502,0,0,0,0,0
36,245282,113,15650228,1173,0,0,0,0,0
37,385976,182,20504906,406,0,0,0,0,0
38,593042,289,26197461,132,0,0,0,0,0
39,891989,595,32678809,27,0,0,0,0,0
40,1314680,1044,39844931,13,0,0,0,0,0
41,1893688,1760,47444000,3,0,0,0,0,0
42,2680168,2949,55207564,2,0,0,0,0,0
43,3714572,4791,62836407,0,0,0,0,0,0
44,5049746,7647,69915593,0,0,0,0,0,0
45,6731759,12411,76143739,0,0,0,0,0,0
46,8805756,19345,81132468,0,0,0,0,0,0
47,11318099,29846,84632172,0,0,0,0,0,0
48,14282539,46147,86429153,0,0,0,0,0,0
49,17720420,70032,86432796,0,0,0,0,0,0
50,21601827,103822,84690330,0,0,0,0,0,0
51,25898845,151751,81271724,0,0,0,0,0,0
52,30557099,219103,76434288,0,0,0,0,0,0
53,35477230,314660,70432506,0,0,0,0,0,0
54,40505818,446267,63643891,0,0,0,0,0,0
55,45556209,621172,56357617,0,0,0,0,0,0
56,50468444,858448,48930437,0,0,0,0,0,0
57,55080683,1168853,41634114,0,0,0,0,0,0
58,59219553,1573525,34739289,0,0,0,0,0,0
59,62767343,2092392,28439314,0,0,0,0,0,0
60,65589702,2753440,22812929,0,0,0,0,0,0
61,67601952,3573865,17950625,0,0,0,0,0,0
62,68708051,4588540,13843752,0,0,0,0,0,0
63,68937309,5821313,10473073,0,0,0,0,0,0
64,68242787,7299416,7772156,0,0,0,0,0,0
65,66665686,9047267,5652778,0,0,0,0,0,0
66,64324238,11089451,4028545,0,0,0,0,0,0
67,61274947,13433359,2815149,0,0,0,0,0,0
68,57647183,16092205,1931062,0,0,0,0,0,0
69,53589215,19039838,1297426,0,0,0,0,0,0
70,49225258,22296841,853731,0,0,0,0,0,0
71,44667866,25821154,551953,0,0,0,0,0,0
72,40095274,29556847,348584,0,0,0,0,0,0
73,35547237,33449276,215866,0,0,0,0,0,0
74,31174943,37456112,130715,0,0,0,0,0,0
75,27035341,41469494,77741,0,0,0,0,0,0
76,23197588,45394555,44748,0,0,0,0,0,0
77,19688109,49151411,25789,0,0,0,0,0,0
78,16534193,52626608,14298,0,0,0,0,0,0
79,13739460,55747545,7812,0,0,0,0,0,0
80,11289373,58379795,4102,0,0,0,0,0,0
81,9195146,60466188,2148,0,0,0,0,0,0
82,7404723,61943403,1107,0,0,0,0,0,0
83,5907424,62788483,563,0,0,0,0,0,0
84,4665583,62918716,272,0,0,0,0,0,0
85,3652072,62377798,116,0,0,0,0,0,0
86,2828956,61135985,61,0,0,0,0,0,0
87,2172622,59280024,19,0,0,0,0,0,0
88,1651878,56847423,13,0,0,0,0,0,0
89,1244766,53917728,4,0,0,0,0,0,0
90,929142,50569619,5,0,0,0,0,0,0
91,686874,46917794,1,0,0,0,0,0,0
92,504855,43051919,0,0,0,0,0,0,0
93,366141,39070126,0,0,0,0,0,0,0
94,264555,35051380,0,0,0,0,0,0,0
95,188991,31112959,0,0,0,0,0,0,0
96,133808,27302641,0,0,0,0,0,0,0
97,93347,23705244,0,0,0,0,0,0,0
98,65072,20350887,0,0,0,0,0,0,0
99,45082,17266328,0,0,0,0,0,0,0
|
|
中级统计结果第二部分
100,30662,14498469,0,0,0,0,0,0,0
101,20959,12029096,0,0,0,0,0,0,0
102,13817,9879472,0,0,0,0,0,0,0
103,9358,8008574,0,0,0,0,0,0,0
104,6275,6431049,0,0,0,0,0,0,0
105,4110,5098652,0,0,0,0,0,0,0
106,2671,4001334,0,0,0,0,0,0,0
107,1712,3106101,0,0,0,0,0,0,0
108,1134,2381018,0,0,0,0,0,0,0
109,719,1803452,0,0,0,0,0,0,0
110,453,1353355,0,0,0,0,0,0,0
111,310,1002428,0,0,0,0,0,0,0
112,218,733294,0,0,0,0,0,0,0
113,109,532057,0,0,0,0,0,0,0
114,71,380667,0,0,0,0,0,0,0
115,40,270080,0,0,0,0,0,0,0
116,24,187551,0,0,0,0,0,0,0
117,25,129932,0,0,0,0,0,0,0
118,9,88952,0,0,0,0,0,0,0
119,6,60341,0,0,0,0,0,0,0
120,3,40224,0,0,0,0,0,0,0
121,2,26671,0,0,0,0,0,0,0
122,1,17207,0,0,0,0,0,0,0
123,0,11029,0,0,0,0,0,0,0
124,1,7106,0,0,0,0,0,0,0
125,0,4379,0,0,0,0,0,0,0
126,0,2853,0,0,0,0,0,0,0
127,0,1784,0,0,0,0,0,0,0
128,1,1062,0,0,0,0,0,0,0
129,0,632,0,0,0,0,0,0,0
130,0,367,0,0,0,0,0,0,0
131,0,241,0,0,0,0,0,0,0
132,0,146,0,0,0,0,0,0,0
133,0,73,0,0,0,0,0,0,0
134,0,41,0,0,0,0,0,0,0
135,0,18,0,0,0,0,0,0,0
136,0,16,0,0,0,0,0,0,0
137,0,11,0,0,0,0,0,0,0
138,0,1,0,0,0,0,0,0,0
139,0,1,0,0,0,0,0,0,0
140,0,1,0,0,0,0,0,0,0
141,0,1,0,0,0,0,0,0,0
142,0,0,0,0,0,0,0,0,0
143,0,0,0,0,0,0,0,0,0
144,0,0,0,0,0,0,0,0,0
145,0,0,0,0,0,0,0,0,0
146,0,0,0,0,0,0,0,0,0
147,0,0,0,0,0,0,0,0,0
148,0,0,0,0,0,0,0,0,0
149,0,0,0,0,0,0,0,0,0
|
|
|
|
