ALL ABOUT BOWLING SCORES

Ever wonder what the most common ten-pin bowling score is? Well, obviously this depends on the skill levels of the players involved, but what if we were to simply look at all possible games and analyze their scores? How many different bowling games are there, and what is the most common score among them? What is the relationship between the score and the average number of strikes and spares? With a little help from the computer, we can come up with some interesting answers. For derivations, see the material further down the page.

For analyses of bowling games other than traditional ten-pin, see

World Bowling (current-frame scoring)
Candlepin & Duckpin Bowling

First, it seems evident that the total number of possible ten-pin bowling games is quite large. We have eleven possibilities for the first ball thrown in the first frame (gutter, 1, 2, ..., 9, strike), and the same possibilities occur for each of the other nine frames. So without even considering the second ball in each frame, at a minimum we have 11¹⁰ = 26 billion possibilities. In fact, the true number of games is much, much larger due to the effect of the second ball in each frame. It's easy to show that the total number of possible games using traditional scoring is

66⁹ x 241 = 5,726,805,883,325,784,576 (about 6 billion billion, or 6 quintillion)

Score Distribution

Calculations of the score distributions for traditionally-scored games results in the following table for the number of possible games and the associated probability if all scores are equally likely:

        Score   Number of possible games   Probability

           0                           1     < 0.01%
           1                          20     < 0.01%
           2                         210     < 0.01%
           3                       1,540     < 0.01%
           4                       8,855     < 0.01%
           5                      42,504     < 0.01%
           6                     177,100     < 0.01%
           7                     657,800     < 0.01%
           8                   2,220,075     < 0.01%
           9                   6,906,900     < 0.01%
          10                  20,030,010     < 0.01%

                              ...

          77     172,542,309,343,731,946       3.01%

                              ...

         288                          12     < 0.01%
         289                          11     < 0.01%
         290                          11     < 0.01%
         291                           1     < 0.01%
         292                           1     < 0.01%
         293                           1     < 0.01%
         294                           1     < 0.01%
         295                           1     < 0.01%
         296                           1     < 0.01%
         297                           1     < 0.01%
         298                           1     < 0.01%
         299                           1     < 0.01%
         300                           1     < 0.01%

The full table is shown below. For each score above above 290, there is only one possible way to play the game. This distribution is shown in the following diagram. It is not precisely symmetric about its maximum point.

The most common score out of all the possibilities is 77. This is the mode of the score distribution. The mean of the distribution is about 79.7, so if you score above that, you can certainly argue that you're doing better than average!

The standard deviation of this score distribution is 13.7 pins.

If we show the cumulative probability histogram corresponding to the score distribution, we can determine the percentiles for various scores:

For example, if we bowl 98 or higher, we're already in the 90th percentile of all possible bowling scores. The median, or 50th percentile, of the score distribution is 79.

Strikes and Spares

The effect of strikes and spares on the average score can be seen in the following contour-line diagram:

As expected, the average score increases with the number of strikes and spares, and the influence of strikes increases with larger scores. It's easy to see from the contour lines in this diagram how much we can expect to increase our average score with more strikes or spares. (The contour lines themselves are only a visual aid since we obviously can't have a fractional number of strikes or spares.) The information in this diagram is presented in tabular form below, along with the minimum and maximum scores possible for various quantities of strikes and spares:

Number of spares
Number of
strikes 0 1 2 3 4 5 6 7 8 9 10
12 300
[300-300]

11 259
[240-299] 277
[270-290]

10 223
[180-288] 240
[210-289] 257
[240-279]

9 193
[120-267] 208
[150-278] 224
[180-279] 239
[210-268]

8 168
[90-246] 181
[120-257] 196
[140-268] 210
[170-268] 224
[200-257]

7 146
[70-225] 159
[90-236] 171
[110-247] 184
[130-257] 198
[150-257] 211
[180-246]

6 128
[60-204] 139
[70-215] 151
[80-226] 163
[100-236] 175
[120-246] 187
[140-246] 199
[170-235]

5 112
[50-183] 123
[60-194] 133
[70-205] 144
[80-215] 155
[90-225] 166
[110-235] 177
[130-235] 188
[150-224]

4 100
[40-162] 109
[50-173] 119
[60-184] 128
[70-194] 138
[80-204] 148
[90-214] 158
[100-224] 168
[120-224] 177
[140-213]

3 89
[30-141] 97
[40-152] 106
[50-163] 115
[60-173] 124
[70-183] 134
[80-193] 143
[90-203] 152
[100-213] 160
[110-213] 166
[130-202]

2 80
[20-120] 88
[30-131] 96
[40-142] 105
[50-152] 114
[60-162] 123
[70-172] 133
[80-182] 142
[90-192] 152
[100-202] 162
[120-202]

1 70
[10-100] 77
[20-111] 85
[30-122] 93
[40-132] 101
[50-142] 110
[60-152] 118
[70-162] 127
[80-172] 136
[90-182] 145
[100-192] 153
[110-192]

0 60
[0-90] 68
[10-100] 75
[20-110] 83
[30-120] 91
[40-130] 99
[50-140] 108
[60-150] 116
[70-160] 126
[80-170] 135
[90-180] 145
[100-190]

	Number of spares
Number of strikes	0	1	2	3	4	5	6	7	8	9	10
12	300 [300-300]
11	259 [240-299]	277 [270-290]
10	223 [180-288]	240 [210-289]	257 [240-279]
9	193 [120-267]	208 [150-278]	224 [180-279]	239 [210-268]
8	168 [90-246]	181 [120-257]	196 [140-268]	210 [170-268]	224 [200-257]
7	146 [70-225]	159 [90-236]	171 [110-247]	184 [130-257]	198 [150-257]	211 [180-246]
6	128 [60-204]	139 [70-215]	151 [80-226]	163 [100-236]	175 [120-246]	187 [140-246]	199 [170-235]
5	112 [50-183]	123 [60-194]	133 [70-205]	144 [80-215]	155 [90-225]	166 [110-235]	177 [130-235]	188 [150-224]
4	100 [40-162]	109 [50-173]	119 [60-184]	128 [70-194]	138 [80-204]	148 [90-214]	158 [100-224]	168 [120-224]	177 [140-213]
3	89 [30-141]	97 [40-152]	106 [50-163]	115 [60-173]	124 [70-183]	134 [80-193]	143 [90-203]	152 [100-213]	160 [110-213]	166 [130-202]
2	80 [20-120]	88 [30-131]	96 [40-142]	105 [50-152]	114 [60-162]	123 [70-172]	133 [80-182]	142 [90-192]	152 [100-202]	162 [120-202]
1	70 [10-100]	77 [20-111]	85 [30-122]	93 [40-132]	101 [50-142]	110 [60-152]	118 [70-162]	127 [80-172]	136 [90-182]	145 [100-192]	153 [110-192]
0	60 [0-90]	68 [10-100]	75 [20-110]	83 [30-120]	91 [40-130]	99 [50-140]	108 [60-150]	116 [70-160]	126 [80-170]	135 [90-180]	145 [100-190]

For example, games with two strikes and four spares average 114 points, but could be as low as 60 or as high as 162, depending on the particular balls thrown in the game.

There are about half a trillion (438,578,270,526) clean games in which all 10 frames are closed, representing only 0.000008% of all possible games.

Results for a Given Score

We can also compute the average number of strikes and spares in a game with a given score:

The average numbers of strikes and spares per game are approximately equal for games scoring about 173. Obviously the only game with 12 strikes (a perfect game) scores 300.

The average number of balls thrown in a game is highly influenced by the number of strikes, but there are side effects resulting from the special scoring in frame 10:

As low scores increase to about 83, the occasional third ball in the tenth frame increases the total number of balls thrown on the average.

The maximum possible number of balls thrown in a game is 21, resulting from nine non-strike frames followed by a tenth frame with three balls.

It's possible to have a game with as few as 11 balls, but the only way would be with all strikes in the first nine frames followed by an open tenth frame. Looking at these 11-ball games in more detail, let A and B be the two balls thrown in frame 10. Since the tenth frame is open, we have 0 ≤ A ≤ 9 (no strike) and 0 ≤ A + B ≤ 9 (no spare). The score of such a game would be 240 + A + 2(A + B):

	Score through frame 7           210
	Points earned in frame 8        20 + A
	Score through frame 8           230 + A
	Points earned in frame 9        10 + A + B
	Score through frame 9           240 + 2A + B
	Points earned in frame 10       A + B
	Final score                     240 + 3A + 2B

The final score for the 11-ball game falls in the range [240,267]. However, the average number of balls thrown for games scoring in this range is well above 11, so the effect of these rare 11-ball games is not apparent in the chart above.

Derivation of the Total Number of Possible Games

In each of frames 1 - 9, there are 66 possible ways to throw the two balls (or single ball in the case of a strike). To see this, let 0 represent a gutter ball and 10 a strike. The possiblities are shown in the following table:

First ball
A Second ball
B Possibilities
Strike
10 (not thrown) 1
9 0 or 1 2
8 0 to 2 3
7 0 to 3 4
6 0 to 4 5
5 0 to 5 6
4 0 to 6 7
3 0 to 7 8
2 0 to 8 9
1 0 to 9 10
0
(gutter) 0 to 10 11
Total 66

First ball A	Second ball B	Possibilities
Strike 10	(not thrown)	1
9	0 or 1	2
8	0 to 2	3
7	0 to 3	4
6	0 to 4	5
5	0 to 5	6
4	0 to 6	7
3	0 to 7	8
2	0 to 8	9
1	0 to 9	10
0 (gutter)	0 to 10	11
Total	66

Since any of the 66 possibilities for the first frame can be followed by any of them for the second frame, and so on, it follows that the number of possible ways to play the first nine frames is 66⁹.

The number of possible outcomes for the two or three balls thrown in Frame 10 can be summarized in the following table:

First ball
A Second ball
B Third ball
C Possibilities Number of
strikes Number of
spares
Strike
10 Strike
10 Strike
10 1 3 0
Non-strike
0 to 9 10 2 0
Non-strike
0 to 9 Spare
10-B 10 1 1
Non-spare
0 to 9-B 55 * 1 0
Non-strike
0 to 9 Spare
10-A Strike
10 10 1 1
Non-strike
0 to 9 100 0 1
Non-spare
0 to 9-A (not thrown) 55 ** 0 0
Total 241

* As B varies from 0 to 9, the sum of the 10-B possibilities 0 to 9-B is:
10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 55
** Same result when A varies varies from 0 to 9

First ball A	Second ball B	Third ball C	Possibilities	Number of strikes	Number of spares
Strike 10	Strike 10	Strike 10	1	3	0
Non-strike 0 to 9	10	2	0
Non-strike 0 to 9	Spare 10-B	10	1	1
Non-spare 0 to 9-B	55 *	1	0
Non-strike 0 to 9	Spare 10-A	Strike 10	10	1	1
Non-strike 0 to 9	100	0	1
Non-spare 0 to 9-A	(not thrown)	55 **	0	0
Total	241

Since there are 241 possible ways to score the tenth frame, the result given at the top of this page for the total number of games follows.

Analysis of Average Strikes and Spares at Higher Scores

As we saw in the above diagram, the average number of strikes and spares exhibits locally non-monotonic behavior at higher scores, probably due to the vastly fewer number of possible games for these scores. For example, the average number of strikes in a 280-point game actually exceeds that in a game scoring 281. This result may run counter to our overall intuition that higher-scoring games should include greater numbers of strikes, but can be confirmed by enumerating the twenty-six possible 280-point games:

                            Frame
Game   1    2    3    4    5    6    7    8    9     10     Strikes

 1.    X    - /  X    X    X    X    X    X    X    X X X     11
 2.    X    1 /  X    X    X    X    X    X    X    X X X     11
 3.    X    2 /  X    X    X    X    X    X    X    X X X     11
 4.    X    3 /  X    X    X    X    X    X    X    X X X     11
 5.    X    4 /  X    X    X    X    X    X    X    X X X     11
 6.    X    5 /  X    X    X    X    X    X    X    X X X     11
 7.    X    6 /  X    X    X    X    X    X    X    X X X     11
 8.    X    7 /  X    X    X    X    X    X    X    X X X     11
 9.    X    8 /  X    X    X    X    X    X    X    X X X     11
10.    X    9 /  X    X    X    X    X    X    X    X X X     11
11.    - /  X    X    X    X    X    X    X    X    X X -     10
12.    1 /  X    X    X    X    X    X    X    X    X X -     10
13.    2 /  X    X    X    X    X    X    X    X    X X -     10
14.    3 /  X    X    X    X    X    X    X    X    X X -     10
15.    4 /  X    X    X    X    X    X    X    X    X X -     10
16.    5 /  X    X    X    X    X    X    X    X    X X -     10
17.    6 /  X    X    X    X    X    X    X    X    X X -     10
18.    7 /  X    X    X    X    X    X    X    X    X X -     10
19.    8 /  X    X    X    X    X    X    X    X    X X -     10
20.    9 /  X    X    X    X    X    X    X    X    X X -     10
21.    X    X    X    X    X    X    X    X    X    X - /     10
22.    X    X    X    X    X    X    X    X    X    X 1 8     10
23.    X    X    X    X    X    X    X    X    X    X 2 6     10
24.    X    X    X    X    X    X    X    X    X    X 3 4     10
25.    X    X    X    X    X    X    X    X    X    X 4 2     10
26.    X    X    X    X    X    X    X    X    X    X 5 -     10

Total                                                        270

and the fifteen possible 281-point games:

                            Frame
Game   1    2    3    4    5    6    7    8    9     10     Strikes

 1.    - /  X    X    X    X    X    X    X    X    X X 1     10
 2.    1 /  X    X    X    X    X    X    X    X    X X 1     10
 3.    2 /  X    X    X    X    X    X    X    X    X X 1     10
 4.    3 /  X    X    X    X    X    X    X    X    X X 1     10
 5.    4 /  X    X    X    X    X    X    X    X    X X 1     10
 6.    5 /  X    X    X    X    X    X    X    X    X X 1     10
 7.    6 /  X    X    X    X    X    X    X    X    X X 1     10
 8.    7 /  X    X    X    X    X    X    X    X    X X 1     10
 9.    8 /  X    X    X    X    X    X    X    X    X X 1     10
10.    9 /  X    X    X    X    X    X    X    X    X X 1     10
11.    X    X    X    X    X    X    X    X    X    X 1 9     10
12.    X    X    X    X    X    X    X    X    X    X 2 7     10
13.    X    X    X    X    X    X    X    X    X    X 3 5     10
14.    X    X    X    X    X    X    X    X    X    X 4 3     10
15.    X    X    X    X    X    X    X    X    X    X 5 1     10

Total                                                        150

The average number of strikes in a 280-point game is 270/26 = 10.38, but the average number in a 281-point game is only 150/15 = 10. A similar type of behavior occurs with the average number of spares.

Ten-pin Score Distribution Table

Here's a full table showing the number of possible games for each traditional score:

Score Number of possible games

0 1

1 20

2 210

3 1540

4 8855

5 42504

6 177100

7 657800

8 2220075

9 6906900

10 20030010

11 54627084

12 141116637

13 347336412

14 818558424

15 1854631380

16 4053948342

17 8574134256

18 17590903116

19 35084425512

20 68153183370

21 129156542039

22 239128282128

23 433093980298

24 768175029950

25 1335679056261

26 2278764308864

27 3817721269708

28 6285424931278

29 10176048813473

30 16210652213304

31 25423690787719

32 39274771758064

33 59789973730461

34 89736657900900

35 132834787033075

36 194006223597572

37 279661205716974

38 398018151390200

39 559449136091831

40 776838931567572

41 1065940588576732

42 1445705502357343

43 1938561121705315

44 2570605432880903

45 3371684590465908

46 4375319099346208

47 5618445228564793

48 7140942201229333

49 8984922304030443

50 11193770355829009

51 13810930667765157

52 16878453276117746

53 20435326129713654

54 24515635362932954

55 29146610869639549

56 34346628376654913

57 40123251227815383

58 46471404549689351

59 53371780703441318

60 60789577452586487

61 68673668434334934

62 76956298564663402

63 85553384395717227

64 94365480254213528

65 103279445170253902

66 112170812747354087

67 120906827121834566

68 129350064451661348

69 137362512979745598

70 144809940796620325

71 151566341291631624

72 157518221668013078

73 162568486673578693

74 166639683923175378

75 169676402232105648

76 171646676234883305

77 172542309343731946

78 172378125687965848

79 171190226627438257

80 169033430825208027

81 165978103316094584

82 162106654714921075

83 157509948809043576

84 152283892386077931

85 146526364181517039

86 140334651650668803

87 133803399444707801

88 127023103852577896

89 120079021507938035

90 113050455155943519

91 106010240661754449

92 99024411737621323

93 92151904402003308

94 85444345654857875

95 78945863453573001

96 72693023944120045

97 66714881583314335

98 61033240145235763

99 55663091133973346

100 50613244155051856

101 45887089510794122

102 41483436078768079

103 37397371704961189

104 33621048067136846

105 30144388614623696

106 26955619314626157

107 24041709119775647

108 21388640692533960

109 18981680119465910

110 16805547548715206

111 14844654231857239

112 13083276623221517

113 11505812292077067

114 10096971927616045

115 8842020009154293

116 7726929590817265

117 6738528470417086

118 5864552560171552

119 5093653838062639

120 4415377510495980

121 3820097597373727

122 3298981687014508

123 2843905747206868

124 2447444695948898

125 2102793053565659

126 1803790254604935

127 1544848145184291

128 1320992367181792

129 1127775864826813

130 961294388171457

131 818085023387881

132 695128788327698

133 589753122859383

134 499630252931260

135 422696870992462

136 357151976811922

137 301400973036441

138 254052574077937

139 213889601295347

140 179862464456172

141 151065169242834

142 126722015973414

143 106169469752641

144 88840622360686

145 74252067274687

146 61990415093876

147 51701385089887

148 43082666091665

149 35870481552300

150 29843343433392

151 24808172866872

152 20607116162379

153 17101443169235

154 14181008701762

155 11747089496422

156 9723545122578

157 8040378083433

158 6644452641044

159 5486702080236

160 4529003381568

161 3736165201688

162 3081105018158

163 2539255963377

164 2091793858275

165 1721930513702

166 1416734360140

167 1164733232308

168 957190045595

169 785911852914

170 645295369580

171 529489941608

172 434606120455

173 356481490646

174 292487050484

175 239755303889

176 196550315542

177 160954253448

178 131791387388

179 107847709116

180 88241591630

181 72162948863

182 59038079745

183 48284335855

184 39509743432

185 32308399043

186 26423428886

187 21582203262

188 17624621529

189 14368737009

190 11720626558

191 9552812749

192 7790240907

193 6351933169

194 5185250585

195 4232118751

196 3457204258

197 2821392492

198 2302090127

199 1874802017

200 1526313637

201 1239515641

202 1007719386

203 818568928

204 666193896

205 542061609

206 442072320

207 360234562

208 293886739

209 239045260

210 194337731

211 157306293

212 127325163

213 102799565

214 83194097

215 67300605

216 54691522

217 44477808

218 36317458

219 29606794

220 24117404

221 19554213

222 15820964

223 12736481

224 10258846

225 8244157

226 6659561

227 5381526

228 4385243

229 3576841

230 2930385

231 2376760

232 1924226

233 1541327

234 1231527

235 975760

236 777090

237 617547

238 498228

239 404981

240 335065

241 275998

242 226966

243 183727

244 148442

245 117291

246 93525

247 73010

248 57960

249 45826

250 37965

251 31193

252 26131

253 21406

254 17422

255 13613

256 10696

257 7975

258 6005

259 4374

260 3534

261 3016

262 2635

263 2264

264 1933

265 1603

266 1323

267 1045

268 810

269 585

270 406

271 277

272 258

273 227

274 206

275 173

276 150

277 115

278 90

279 53

280 26

281 15

282 15

283 14

284 14

285 13

286 13

287 12

288 12

289 11

290 11

291 1

292 1

293 1

294 1

295 1

296 1

297 1

298 1

299 1

300 1

Total 5726805883325784576

Score	Number of possible games
0	1
1	20
2	210
3	1540
4	8855
5	42504
6	177100
7	657800
8	2220075
9	6906900
10	20030010
11	54627084
12	141116637
13	347336412
14	818558424
15	1854631380
16	4053948342
17	8574134256
18	17590903116
19	35084425512
20	68153183370
21	129156542039
22	239128282128
23	433093980298
24	768175029950
25	1335679056261
26	2278764308864
27	3817721269708
28	6285424931278
29	10176048813473
30	16210652213304
31	25423690787719
32	39274771758064
33	59789973730461
34	89736657900900
35	132834787033075
36	194006223597572
37	279661205716974
38	398018151390200
39	559449136091831
40	776838931567572
41	1065940588576732
42	1445705502357343
43	1938561121705315
44	2570605432880903
45	3371684590465908
46	4375319099346208
47	5618445228564793
48	7140942201229333
49	8984922304030443
50	11193770355829009
51	13810930667765157
52	16878453276117746
53	20435326129713654
54	24515635362932954
55	29146610869639549
56	34346628376654913
57	40123251227815383
58	46471404549689351
59	53371780703441318
60	60789577452586487
61	68673668434334934
62	76956298564663402
63	85553384395717227
64	94365480254213528
65	103279445170253902
66	112170812747354087
67	120906827121834566
68	129350064451661348
69	137362512979745598
70	144809940796620325
71	151566341291631624
72	157518221668013078
73	162568486673578693
74	166639683923175378
75	169676402232105648
76	171646676234883305
77	172542309343731946
78	172378125687965848
79	171190226627438257
80	169033430825208027
81	165978103316094584
82	162106654714921075
83	157509948809043576
84	152283892386077931
85	146526364181517039
86	140334651650668803
87	133803399444707801
88	127023103852577896
89	120079021507938035
90	113050455155943519
91	106010240661754449
92	99024411737621323
93	92151904402003308
94	85444345654857875
95	78945863453573001
96	72693023944120045
97	66714881583314335
98	61033240145235763
99	55663091133973346
100	50613244155051856
101	45887089510794122
102	41483436078768079
103	37397371704961189
104	33621048067136846
105	30144388614623696
106	26955619314626157
107	24041709119775647
108	21388640692533960
109	18981680119465910
110	16805547548715206
111	14844654231857239
112	13083276623221517
113	11505812292077067
114	10096971927616045
115	8842020009154293
116	7726929590817265
117	6738528470417086
118	5864552560171552
119	5093653838062639
120	4415377510495980
121	3820097597373727
122	3298981687014508
123	2843905747206868
124	2447444695948898
125	2102793053565659
126	1803790254604935
127	1544848145184291
128	1320992367181792
129	1127775864826813
130	961294388171457
131	818085023387881
132	695128788327698
133	589753122859383
134	499630252931260
135	422696870992462
136	357151976811922
137	301400973036441
138	254052574077937
139	213889601295347
140	179862464456172
141	151065169242834
142	126722015973414
143	106169469752641
144	88840622360686
145	74252067274687
146	61990415093876
147	51701385089887
148	43082666091665
149	35870481552300
150	29843343433392
151	24808172866872
152	20607116162379
153	17101443169235
154	14181008701762
155	11747089496422
156	9723545122578
157	8040378083433
158	6644452641044
159	5486702080236
160	4529003381568
161	3736165201688
162	3081105018158
163	2539255963377
164	2091793858275
165	1721930513702
166	1416734360140
167	1164733232308
168	957190045595
169	785911852914
170	645295369580
171	529489941608
172	434606120455
173	356481490646
174	292487050484
175	239755303889
176	196550315542
177	160954253448
178	131791387388
179	107847709116
180	88241591630
181	72162948863
182	59038079745
183	48284335855
184	39509743432
185	32308399043
186	26423428886
187	21582203262
188	17624621529
189	14368737009
190	11720626558
191	9552812749
192	7790240907
193	6351933169
194	5185250585
195	4232118751
196	3457204258
197	2821392492
198	2302090127
199	1874802017
200	1526313637
201	1239515641
202	1007719386
203	818568928
204	666193896
205	542061609
206	442072320
207	360234562
208	293886739
209	239045260
210	194337731
211	157306293
212	127325163
213	102799565
214	83194097
215	67300605
216	54691522
217	44477808
218	36317458
219	29606794
220	24117404
221	19554213
222	15820964
223	12736481
224	10258846
225	8244157
226	6659561
227	5381526
228	4385243
229	3576841
230	2930385
231	2376760
232	1924226
233	1541327
234	1231527
235	975760
236	777090
237	617547
238	498228
239	404981
240	335065
241	275998
242	226966
243	183727
244	148442
245	117291
246	93525
247	73010
248	57960
249	45826
250	37965
251	31193
252	26131
253	21406
254	17422
255	13613
256	10696
257	7975
258	6005
259	4374
260	3534
261	3016
262	2635
263	2264
264	1933
265	1603
266	1323
267	1045
268	810
269	585
270	406
271	277
272	258
273	227
274	206
275	173
276	150
277	115
278	90
279	53
280	26
281	15
282	15
283	14
284	14
285	13
286	13
287	12
288	12
289	11
290	11
291	1
292	1
293	1
294	1
295	1
296	1
297	1
298	1
299	1
300	1
Total	5726805883325784576

Computation Algorithm

Obviously it isn't feasible to score all 6 quintillion possible games and enumerate the results individually. But with a "divide-and-conquer" strategy, we can derive the correct results by performing convolutions on small portions of the game. The convolution methodology is based on combining all possible types of throwing patterns from one group of frames with all possible patterns in a subsequent group of frames. The quantity stored in memory arrays is the number of games with the given combination of parameters, so the resulting convolution is formed simply by multiplying the number of games in the first group by the number of games in the second group since any game in the first group can be followed by any of the games in the second group. The total number of games is less than 2⁶³ and so can be stored in an 8-byte signed integer data type.

The first group of frames must track four scoring possibilities forward that require knowledge of the first two balls A and B thrown in the second group of frames. These four possibilities are: 1) an open frame resulting in no extra score (0 points added); 2) a spare adding in the score from the next ball (A); 3) a strike adding in the score from the next two balls (A + B); and 4) a double strike in the last two frames adding in twice the score from the next ball as well as the score from the second ball following (2A + B). Results are kept in four separate tables for each partial score. To be properly combined in the convolution, the array in the second group is identified by all possible values for its first two balls A and B, which are tracked backward into the convolution performed with the first group of frames. To simplify programming, A and B are each allowed to range from 0 to 10 (121 total combinations) even though certain combinations of A and B are invalid (these invalid combinations will have zero entries for the number of games and are ignored in the computations).

An array of possibilities for the first three frames is computed and then convolved with itself to form results for the first six frames. This result is convolved with the array for the first three frames to produce results for the first nine frames. Finally, that result is convolved with computations for frame 10 in order to generate results corresponding to the entire game.

World Bowling (Current-Frame) Scoring

An alternative and simpler approach to scoring bowling games is the current-frame scoring method supported by the World Bowling organization. In this method, only the knowledge of the balls in the currently-bowled frame is necessary to determine the score; there is no dependency on future balls to be thrown, and there are no bonus balls in Frame 10. Thus, the score possibilities are independent and identically distributed for every frame. Current-frame scoring is defined as follows:

A strike earns 30 points for that frame
A spare earns 10 points plus the number of pins for the first ball
An open frame earns the total of both balls thrown, as in traditional scoring

Therefore, there are 21 possible scores for one frame: 0,1,...,9 (open frames), 10,11,...,19 (spares), 30 (strike). It's not possible to generate a one-frame score between 20 and 29, inclusive. Adding up the possible scores for 10 frames generates a game total between 0 and 300, except for the impossible range 290-299. As in traditional scoring, there is only one perfect game scoring the maximum of 300 points, but it is bowled by throwing 10, rather than 12, consecutive strikes. The lack of bonus balls has a significant effect on the current-frame scoring statistics, as compared to traditional scoring.

As derived in the table above, there again are 66 possible ways to bowl a single frame. Since all ten frames have the same possibilities, the total number of possible games is

66¹⁰ = 1,568,336,880,910,795,776 = 1.6 x 10¹⁸ approximately

This is about one-quarter (66/241) of the number of different games with traditional ten-pin bowling scoring.

Single-Frame Analysis

The possible scores for a single frame are shown in the following table:

Single-
frame
scores First Ball
0 1 2 3 4 5 6 7 8 9 10
Second
Ball 0 0 1 2 3 4 5 6 7 8 9 30
1 1 2 3 4 5 6 7 8 9 19
2 2 3 4 5 6 7 8 9 18
3 3 4 5 6 7 8 9 17
4 4 5 6 7 8 9 16
5 5 6 7 8 9 15
6 6 7 8 9 14
7 7 8 9 13
8 8 9 12
9 9 11
10 10

Single- frame scores	First Ball
0	1	2	3	4	5	6	7	8	9	10
Second Ball	0	0	1	2	3	4	5	6	7	8	9	30
1	1	2	3	4	5	6	7	8	9	19
2	2	3	4	5	6	7	8	9	18
3	3	4	5	6	7	8	9	17
4	4	5	6	7	8	9	16
5	5	6	7	8	9	15
6	6	7	8	9	14
7	7	8	9	13
8	8	9	12
9	9	11
10	10

If all scores are considered to be equally likely, we can count up the results in the preceding table to determine the probability p(s) of one frame's score being s:

	       { (s+1)/66, 0 ≤ s ≤ 9 (open frame)
	       {
	       { 1/66,     10 ≤ s ≤ 19 (spare)
	p(s) = {
	       { 1/66,     s = 30 (strike)
	       {
	       { 0,        otherwise

The mean of this score distribution is 505/66 = 7.65 pins. The probability of a strike is 1/66 = 1.515%, and the probability of a spare is ten times that, or 15.15%. The probability of an open frame is 55/66 = 83.33%. A histogram of the score distribution is shown below, alongside a comparable plot for the scoring distribution of the first frame using traditional scoring:

Statistical Analysis

At first glance, current-frame scoring could be expected to increase average scores over all possible games since every strike earns as many points as the first strike of a turkey in traditional scoring. This is borne out by the statistics below for the first frame:

First frame Tenth frame Game
World Traditional World Traditional World Traditional
Mode 9 9 9 20 72 77
Mean 7.65 7.42 7.65 13.84 76.5 79.7
Median 7 7 7 15 75 79

		First frame		Tenth frame		Game
World	Traditional	World	Traditional	World	Traditional
Mode	9	9	9	20	72	77
Mean	7.65	7.42	7.65	13.84	76.5	79.7
Median	7	7	7	15	75	79

However, as seen in the table above, the score statistics for the overall game using World Bowling (current-frame) scoring are slightly lower. This is due the effect of the bonus balls and associated tenth-frame scoring in the traditional game, shown below:

This effect skews the overall results for the entire World Bowling game, making lower scores slightly more likely with current-frame scoring:

Here, the asymmetry of the probability distribution about its maximum point (mode) is more apparent than with traditional scoring. The standard deviation of this score distribution is 15.2 pins. The full table of possible scores and frequencies is shown below.

The corresponding cumulative probability histogram shows the score percentiles, all of which are lower than with traditional scoring:

The average number of strikes or spares needed to reach a certain score is generally less than with traditional scoring. Conversely, the average score associated with a given number of strikes and spares can be signfiicantly higher.

Results for a Given Score

The average number of strikes and spares in a game with a given score is as follows:

Results are omitted for the one (perfect) game scoring above 290. The average numbers of strikes and spares per game are approximately the same for games scoring 166. The number of closed frames in a game is simply the number of strikes plus the number of spares, since there are no bonus balls in Frame 10. The average number of spares has a locally non-monotonic or "scalloped" shape at higher scores due to the effects of the score gap between spares and a strike.

The number of balls thrown in a game is 20 minus the number of strikes, and ranges from 11 to 20 for games scoring other than 300:

Results for Given Numbers of Strikes and Spares

For a given number of strikes T and number of spares P, the minimum and maximum game score can be determined in closed form as follows: The portion of the game score from strikes is fixed at 30T. The minimum-scoring spare (10) occurs when a gutter ball is followed by knocking down all ten pins, and the maximum-scoring spare occurs when a 9-pin ball is followed by a 1-pin ball (score of 19). The minimum score for an open frame is 0, occuring when two gutter balls are thrown, and the maximum score for an open frame is 9, occurring when the pinfalls for the two balls add up to 9. Note that in the case of both a spare and an open frame, the spread in scores for that frame is 9. Therefore, we have the following table:

T strikes P spares 10-T-P open frames Game total
Minimum score 30T 10P 0 30T + 10P
Maximum score 30T 19P 9(10-T-P) 21T + 10P + 90

	T strikes	P spares	10-T-P open frames	Game total
Minimum score	30T	10P	0	30T + 10P
Maximum score	30T	19P	9(10-T-P)	21T + 10P + 90

The spread between the minimum and maximum game scores is 9(10-T), which is the sum of the spreads for all the non-strike frames. The average, minimum and maximum scores possible for various quantities of strikes and spares is shown in the table below, along with the change in average score from traditional scoring:

Number of spares
Number of
strikes 0 1 2 3 4 5 6 7 8 9 10
10 300
(+77)
[300-300]

9 276
(+83)
[270-279] 285
(+77)
[280-289]

8 252
(+84)
[240-258] 261
(+80)
[250-268] 269
(+73)
[260-278]

7 228
(+82)
[210-237] 237
(+78)
[220-247] 245
(+74)
[230-257] 254
(+70)
[240-267]

6 204
(+76)
[180-216] 213
(+74)
[190-226] 221
(+70)
[200-236] 230
(+67)
[210-246] 238
(+63)
[220-256]

5 180
(+68)
[150-195] 189
(+66)
[160-205] 197
(+64)
[170-215] 206
(+62)
[180-225] 214
(+59)
[190-235] 223
(+57)
[200-245]

4 156
(+56)
[120-174] 165
(+56)
[130-184] 173
(+54)
[140-194] 182
(+54)
[150-204] 190
(+52)
[160-214] 199
(+51)
[170-224] 207
(+49)
[180-234]

3 132
(+43)
[90-153] 141
(+44)
[100-163] 149
(+43)
[110-173] 158
(+43)
[120-183] 166
(+42)
[130-193] 175
(+41)
[140-203] 183
(+40)
[150-213] 192
(+40)
[160-223]

2 108
(+28)
[60-132] 117
(+29)
[70-142] 125
(+29)
[80-152] 133
(+28)
[90-162] 142
(+28)
[100-172] 151
(+28)
[110-182] 159
(+26)
[120-192] 168
(+26)
[130-202] 176
(+24)
[140-212]

1 84
(+14)
[30-111] 92
(+15)
[40-121] 101
(+16)
[50-131] 109
(+16)
[60-141] 118
(+17)
[70-151] 126
(+16)
[80-161] 135
(+17)
[90-171] 144
(+17)
[100-181] 152
(+16)
[110-191] 161
(+16)
[120-201]

0 60
(0)
[0-90] 69
(+1)
[10-100] 77
(+2)
[20-110] 85
(+2)
[30-120] 94
(+3)
[40-130] 103
(+4)
[50-140] 111
(+3)
[60-150] 120
(+4)
[70-160] 128
(+2)
[80-170] 137
(+2)
[90-180] 145
(0)
[100-190]

	Number of spares
Number of strikes	0	1	2	3	4	5	6	7	8	9	10
10	300 (+77) [300-300]
9	276 (+83) [270-279]	285 (+77) [280-289]
8	252 (+84) [240-258]	261 (+80) [250-268]	269 (+73) [260-278]
7	228 (+82) [210-237]	237 (+78) [220-247]	245 (+74) [230-257]	254 (+70) [240-267]
6	204 (+76) [180-216]	213 (+74) [190-226]	221 (+70) [200-236]	230 (+67) [210-246]	238 (+63) [220-256]
5	180 (+68) [150-195]	189 (+66) [160-205]	197 (+64) [170-215]	206 (+62) [180-225]	214 (+59) [190-235]	223 (+57) [200-245]
4	156 (+56) [120-174]	165 (+56) [130-184]	173 (+54) [140-194]	182 (+54) [150-204]	190 (+52) [160-214]	199 (+51) [170-224]	207 (+49) [180-234]
3	132 (+43) [90-153]	141 (+44) [100-163]	149 (+43) [110-173]	158 (+43) [120-183]	166 (+42) [130-193]	175 (+41) [140-203]	183 (+40) [150-213]	192 (+40) [160-223]
2	108 (+28) [60-132]	117 (+29) [70-142]	125 (+29) [80-152]	133 (+28) [90-162]	142 (+28) [100-172]	151 (+28) [110-182]	159 (+26) [120-192]	168 (+26) [130-202]	176 (+24) [140-212]
1	84 (+14) [30-111]	92 (+15) [40-121]	101 (+16) [50-131]	109 (+16) [60-141]	118 (+17) [70-151]	126 (+16) [80-161]	135 (+17) [90-171]	144 (+17) [100-181]	152 (+16) [110-191]	161 (+16) [120-201]
0	60 (0) [0-90]	69 (+1) [10-100]	77 (+2) [20-110]	85 (+2) [30-120]	94 (+3) [40-130]	103 (+4) [50-140]	111 (+3) [60-150]	120 (+4) [70-160]	128 (+2) [80-170]	137 (+2) [90-180]	145 (0) [100-190]

Probability Distribution of Strikes and Spares

If we set aside scoring for the moment and continue considering all possible ball throws as equally likely, then the number of strikes T and number of spares P in a game follows the trinomial probability distribution

	               10!             T        P       10-T-P
	g(T,P) = --------------- (1/66)  (10/66) (55/66)
	         T! P! (10-T-P)!

	      for 0 ≤ T ≤ 10 and 0 ≤ P ≤ 10-T

which is based on the one-frame probabilities defined above for a strike, a spare or an open frame. For example, the fraction of all possible games that have exactly 1 strike and 1 spare is

	           10!                          8
	g(1,1) = -------- (1/66) (10/66) (55/66)  = 4.81%
	         1! 1! 8!

Evaluating these results for the feasible ranges of strikes and spares, we have the following plot:

The frequencies associated with 3 or more strikes are too low to show up on the chart.

Here's the same information in the form of a 3D histogram:

Clean Games

As we saw earlier, the number of ways of getting a strike or a spare in one frame is 1 + 10 = 11. It follows that the number of ways of obtaining 10 closed frames is

11¹⁰ = 25,937,424,601

so there are about 26 billion clean games in which all 10 frames are closed, representing only 0.0000016% of all possible games. Obviously, no game with a score under 100 can be a clean game since a strike earns more than any spare and 10 spares must earn at least 100 points.

Games scoring 269, 280-289, or 300 must always be clean games:

Any game scores more points than any other game that has fewer strikes.
A 269-point game has 8 strikes and 2 spares:
- 9-strike game score ≥ 9 strikes + 1 open frame ≥ 9x30 + 0 = 270, which is too large.
- 7-strike game score ≤ 7 strikes + 3 spares ≤ 7x30 + 3x19 = 267, which is too small.
- Therefore there are exactly 8 strikes in a 269-point game.
- 0-spare game score = 8 strikes + 2 open frames ≤ 8x30 + 2x9 = 258, which is too small.
- 1-spare game score = 8 strikes + 1 spare + 1 open frame ≤ 8x30 + 19 + 9 = 268, which is too small.
- 2-spare game score is in the range [8x30 + 2x10, 8x30 + 2x19] = [260,278], which includes 269.
- Therefore there are exactly 2 spares in every 269-point game.
Games scoring 280-289 have 9 strikes and 1 spare:
- 10-strike game score = 10x30 = 300, which is too large.
- 8-strike game score ≤ 8 strikes + 2 spares ≤ 8x30 + 2x19 = 278, which is too small.
- Therefore there are exactly 9 strikes in a game scoring 280-289 points.
- 0-spare game score = 9 strikes + 1 open frame ≤ 9x30 + 9 = 279, which is too small.
- 1-spare game score is in the range [9x30 + 10, 9x30 + 19] = [280,289], which is the specified range.
- Therefore there is exactly 1 spare in every game scoring 280-289 points.

Effect of the Strike Score

Let M be the number of points assigned to a strike; previously we have been examining only the case M = 30. When M ≥ 21, there can be no games scoring between 9M + 20 and 10M - 1, inclusive, as shown by the following analysis:

10-strike game score = 10M, which is larger than any score in the range 9M + 20 to 10M - 1.
9-strike game score ≤ 9 strikes + 1 spare ≤ 9M + 19, which is smaller than any score in the range 9M + 20 to 10M - 1.

For example, in the usual World Bowling scoring with M = 30, the impossible range of scores is 290 to 299. When M = 21, this reduces to just the single impossible score 209, and when M = 20, there are no gaps in the game scoring range 0 - 200.

Analyses similar to those in the preceding sections show that the statistics of the game score distribution do not vary greatly for M < 30:

Game score statistic
Mean Mode Median
Strike
score 20 75.0 73 74
25 75.8 72 75
30 76.5 72 75

	Game score statistic
Mean	Mode	Median
Strike score	20	75.0	73	74
25	75.8	72	75
30	76.5	72	75

However, both the average number of strikes and spares for a given score, and the average score for a given number of strikes and spares can vary widely. In the former case, some of the irregular behavior of the average number of spares at higher scores may be eliminated. For example, the left-hand plot below shows results when 25 points are assigned to a strike instead of 30, and the right-hand plot shows comparable results when a strike earns only 20 points:

World Bowling Score Distribution Table

Here's a full table showing the number of possible games for each score in World Bowling:

Score Number of possible games

0 1

1 20

2 210

3 1540

4 8855

5 42504

6 177100

7 657800

8 2220075

9 6906900

10 20029910

11 54625390

12 141101325

13 347238500

14 818062300

15 1852514070

16 4046044310

17 8547655600

18 17509826400

19 34854380550

20 67541888900

21 127622461010

22 235467859445

23 424744086650

24 749883201650

25 1297052375366

26 2199889956935

27 3661566327540

28 5985003339040

29 9613307642730

30 15182582176769

31 23589203727170

32 36073308835395

33 54319527255130

34 80574860327825

35 117781961461410

36 169723917774345

37 241173996359780

38 338040825065020

39 467496338531750

40 638070844965110

41 859697155357480

42 1143684325691265

43 1502601676134770

44 1950055813772110

45 2500347731015040

46 3168003824781765

47 3967183760495780

48 4910979046898705

49 6010628213027240

50 7274686502965781

51 8708198717340170

52 10311931823519525

53 12081727884792140

54 14008036633963050

55 16075680003223462

56 18263888111000370

57 20546628328775240

58 22893227578599125

59 25269264984684270

60 27637689796314787

61 29960100551745920

62 32198107959174615

63 34314697579840260

64 36275509938490805

65 38049964968674380

66 39612173449501415

67 40941598074127430

68 42023448224326255

69 42848812855238280

70 43414552998372050

71 43722987768421870

72 43781414679391290

73 43601506663391340

74 43198625417862815

75 42591085277769086

76 41799395825565305

77 40845506909141420

78 39752077718176655

79 38541791092813400

80 37236734265954227

81 35857866748978560

82 34424594142676465

83 32954462581014750

84 31462981911416615

85 29963576660029292

86 28467653020803480

87 26984759038298590

88 25522806327282305

89 24088318835376660

90 22686676709427605

91 21322330217135730

92 19998968445645700

93 18719638387834240

94 17486820124802685

95 16302471277776220

96 15168057254583800

97 14084582319246780

98 13052630658586615

99 12072418722804900

100 11143854188694724

101 10266593344594420

102 9440087767430720

103 8663612863528440

104 7936274875808590

105 7256998619048336

106 6624504282096975

107 6037286287205860

108 5493607797089085

109 4991517434455090

110 4528888417358896

111 4103475135395670

112 3712978649550105

113 3355111029457150

114 3027649006567760

115 2728470033340074

116 2455568079499915

117 2207051522130810

118 1981129914883315

119 1776098207280280

120 1590324559115145

121 1422245289812010

122 1270367906237890

123 1133280782366540

124 1009666167227130

125 898312049098324

126 798118306584825

127 708093859220590

128 627344542961095

129 555056550996240

130 490479529459266

131 432912437384680

132 381694130477375

133 336199390618620

134 295839903993625

135 260068624420566

136 228385209907195

137 200339988923560

138 175534435457010

139 153617687340420

140 134280575449174

141 117248552344640

142 102274752682935

143 89134183607230

144 77619732640965

145 67540296613840

146 58720883516950

147 51004028108040

148 44251302626970

149 38343110315140

150 33177167222081

151 28666108631730

152 24734664812935

153 21316832602560

154 18353420191210

155 15790257735412

156 13577241099990

157 11668204865040

158 10021398119875

159 8600056590950

160 7372547220108

161 6312149354020

162 5396549489370

163 4607137627000

164 3928202327005

165 3346128105804

166 2848702430145

167 2424639770410

168 2063426478455

169 1755582128400

170 1492817304730

171 1268092189410

172 1075584180220

173 910576986280

174 769288201635

175 648657261636

176 546120947470

177 459409229300

178 386400235770

179 325079514760

180 273596522207

181 230305907930

182 193793062990

183 162883715630

184 136637715655

185 114327530998

186 95402395945

187 79439498830

188 66084078730

189 54980816560

190 45786537349

191 38181246870

192 31877185060

193 26625569460

194 22220692830

195 18501030150

196 15347001405

197 12675027870

198 10427511060

199 8558355090

200 7019752752

201 5763798430

202 4744025670

203 3916846800

204 3242871225

205 2688078252

206 2224819530

207 1832625420

208 1498788840

209 1218699360

210 987127782

211 798400920

212 646587315

213 525693840

214 429873150

215 353641932

216 292109910

217 241219560

218 197996490

219 160810440

220 129328302

221 103144140

222 81787560

223 64732620

224 51407280

225 41203392

226 33487230

227 27610560

228 22922250

229 18780420

230 15177312

231 12097680

232 9518700

233 7409880

234 5732970

235 4441872

236 3482550

237 2792940

238 2302860

239 1933920

240 1599447

241 1300020

242 1035990

243 807480

244 614385

245 456372

246 332880

247 243120

248 186075

249 160500

250 136530

251 114270

252 93825

253 75300

254 58800

255 44430

256 32295

257 22500

258 15150

259 10350

260 9225

261 8130

262 7065

263 6030

264 5025

265 4050

266 3105

267 2190

268 1305

269 450

270 415

271 380

272 345

273 310

274 275

275 240

276 205

277 170

278 135

279 100

280 10

281 10

282 10

283 10

284 10

285 10

286 10

287 10

288 10

289 10

290 0

291 0

292 0

293 0

294 0

295 0

296 0

297 0

298 0

299 0

300 1

Total 1568336880910795776

Score	Number of possible games
0	1
1	20
2	210
3	1540
4	8855
5	42504
6	177100
7	657800
8	2220075
9	6906900
10	20029910
11	54625390
12	141101325
13	347238500
14	818062300
15	1852514070
16	4046044310
17	8547655600
18	17509826400
19	34854380550
20	67541888900
21	127622461010
22	235467859445
23	424744086650
24	749883201650
25	1297052375366
26	2199889956935
27	3661566327540
28	5985003339040
29	9613307642730
30	15182582176769
31	23589203727170
32	36073308835395
33	54319527255130
34	80574860327825
35	117781961461410
36	169723917774345
37	241173996359780
38	338040825065020
39	467496338531750
40	638070844965110
41	859697155357480
42	1143684325691265
43	1502601676134770
44	1950055813772110
45	2500347731015040
46	3168003824781765
47	3967183760495780
48	4910979046898705
49	6010628213027240
50	7274686502965781
51	8708198717340170
52	10311931823519525
53	12081727884792140
54	14008036633963050
55	16075680003223462
56	18263888111000370
57	20546628328775240
58	22893227578599125
59	25269264984684270
60	27637689796314787
61	29960100551745920
62	32198107959174615
63	34314697579840260
64	36275509938490805
65	38049964968674380
66	39612173449501415
67	40941598074127430
68	42023448224326255
69	42848812855238280
70	43414552998372050
71	43722987768421870
72	43781414679391290
73	43601506663391340
74	43198625417862815
75	42591085277769086
76	41799395825565305
77	40845506909141420
78	39752077718176655
79	38541791092813400
80	37236734265954227
81	35857866748978560
82	34424594142676465
83	32954462581014750
84	31462981911416615
85	29963576660029292
86	28467653020803480
87	26984759038298590
88	25522806327282305
89	24088318835376660
90	22686676709427605
91	21322330217135730
92	19998968445645700
93	18719638387834240
94	17486820124802685
95	16302471277776220
96	15168057254583800
97	14084582319246780
98	13052630658586615
99	12072418722804900
100	11143854188694724
101	10266593344594420
102	9440087767430720
103	8663612863528440
104	7936274875808590
105	7256998619048336
106	6624504282096975
107	6037286287205860
108	5493607797089085
109	4991517434455090
110	4528888417358896
111	4103475135395670
112	3712978649550105
113	3355111029457150
114	3027649006567760
115	2728470033340074
116	2455568079499915
117	2207051522130810
118	1981129914883315
119	1776098207280280
120	1590324559115145
121	1422245289812010
122	1270367906237890
123	1133280782366540
124	1009666167227130
125	898312049098324
126	798118306584825
127	708093859220590
128	627344542961095
129	555056550996240
130	490479529459266
131	432912437384680
132	381694130477375
133	336199390618620
134	295839903993625
135	260068624420566
136	228385209907195
137	200339988923560
138	175534435457010
139	153617687340420
140	134280575449174
141	117248552344640
142	102274752682935
143	89134183607230
144	77619732640965
145	67540296613840
146	58720883516950
147	51004028108040
148	44251302626970
149	38343110315140
150	33177167222081
151	28666108631730
152	24734664812935
153	21316832602560
154	18353420191210
155	15790257735412
156	13577241099990
157	11668204865040
158	10021398119875
159	8600056590950
160	7372547220108
161	6312149354020
162	5396549489370
163	4607137627000
164	3928202327005
165	3346128105804
166	2848702430145
167	2424639770410
168	2063426478455
169	1755582128400
170	1492817304730
171	1268092189410
172	1075584180220
173	910576986280
174	769288201635
175	648657261636
176	546120947470
177	459409229300
178	386400235770
179	325079514760
180	273596522207
181	230305907930
182	193793062990
183	162883715630
184	136637715655
185	114327530998
186	95402395945
187	79439498830
188	66084078730
189	54980816560
190	45786537349
191	38181246870
192	31877185060
193	26625569460
194	22220692830
195	18501030150
196	15347001405
197	12675027870
198	10427511060
199	8558355090
200	7019752752
201	5763798430
202	4744025670
203	3916846800
204	3242871225
205	2688078252
206	2224819530
207	1832625420
208	1498788840
209	1218699360
210	987127782
211	798400920
212	646587315
213	525693840
214	429873150
215	353641932
216	292109910
217	241219560
218	197996490
219	160810440
220	129328302
221	103144140
222	81787560
223	64732620
224	51407280
225	41203392
226	33487230
227	27610560
228	22922250
229	18780420
230	15177312
231	12097680
232	9518700
233	7409880
234	5732970
235	4441872
236	3482550
237	2792940
238	2302860
239	1933920
240	1599447
241	1300020
242	1035990
243	807480
244	614385
245	456372
246	332880
247	243120
248	186075
249	160500
250	136530
251	114270
252	93825
253	75300
254	58800
255	44430
256	32295
257	22500
258	15150
259	10350
260	9225
261	8130
262	7065
263	6030
264	5025
265	4050
266	3105
267	2190
268	1305
269	450
270	415
271	380
272	345
273	310
274	275
275	240
276	205
277	170
278	135
279	100
280	10
281	10
282	10
283	10
284	10
285	10
286	10
287	10
288	10
289	10
290	0
291	0
292	0
293	0
294	0
295	0
296	0
297	0
298	0
299	0
300	1
Total	1568336880910795776

Candlepin & Duckpin Bowling

Less commonly played games include candlepin and duckpin bowling. Although the pins and balls are sized differently between these two games (see diagram), their scoring is identical. The main gameplay difference is that in candlepin bowling, fallen pins are not cleared away between the balls of a frame. Ten pins and ten scoring frames (boxes) are used as in ten-pin bowling, as well as the usual bonus points consisting of the next ball for a spare and the next two balls for a strike. Extra balls can be rolled in the tenth frame as in ten-pin bowling.

The only difference in scoring between candlepin/duckpin and ten-pin bowling is that a third ball is available in each of the first nine frames when neither a strike nor a spare is thrown. No bonus points beyond the pin count are awarded for the third ball in a frame, so the maximum score for an open frame is 10 (vs. 9 in ten-pin bowling). Three balls are always thrown in the last frame. The third balls present in lower-scoring candlepin and duckpin games increases the total number of possibilities to about a million times that of ten-pin bowling:

286⁹ x 461 = 5,901,945,890,697,151,108,875,776 = 5.9 x 10²⁴ approximately

See the section below for a derivation.

Score Distribution

The distribution of scores if all games are equally likely is shown in the following diagram:

The mode (most common score) of the distribution is 79 and the average score is about 79.4, which are both very nearly the same as in ten-pin bowling. The standard deviation of the score distribution is 9.3, which is considerably less than in ten-pin bowling and can be seen in the narrower score distribution that has less-numerous higher and lower scores. The full table of possible scores and frequencies is shown below.

The corresponding cumulative probability histogram shows the percentiles for various scores:

For example, if we bowl 91 or higher, we're in the 90th percentile of all possible bowling scores. The median (50th percentile) of the score distribution is 79, the same as in ten-pin bowling.

Results for a Given Score

The average number of strikes and spares in a game with a given score is as follows:

The average numbers of strikes and spares per game are approximately the same for games scoring 117.

The number of balls thrown in a game ranges from 12 (all strikes in the first 9 frames) to 30 (first 9 frames all open):

Strikes and Spares

The average, minimum and maximum scores possible for various quantities of strikes and spares is shown in the table below, along with the change in average score from traditional scoring:

Number of spares
Number of
strikes 0 1 2 3 4 5 6 7 8 9 10
12 300
(0)
[300-300]
11 258
(-1)
[240-299] 277
(0)
[270-290]
10 223
(0)
[180-288] 239
(-1)
[210-289] 257
(0)
[240-280]
9 193
(0)
[120-268] 208
(0)
[150-278] 223
(-1)
[180-279] 239
(0)
[210-269]
8 168
(0)
[90-248] 181
(0)
[110-258] 195
(-1)
[130-268] 209
(-1)
[160-268] 224
(0)
[190-258]
7 147
(+1)
[70-228] 159
(0)
[80-238] 171
(0)
[100-248] 184
(0)
[120-257] 197
(-1)
[150-257] 210
(-1)
[180-247]
6 131
(+3)
[60-208] 141
(+2)
[60-218] 151
(0)
[70-228] 162
(-1)
[90-237] 173
(-2)
[110-246] 185
(-2)
[130-246] 198
(-1)
[160-236]
5 117
(+5)
[50-188] 125
(+2)
[50-198] 134
(+1)
[60-208] 143
(-1)
[70-217] 153
(-2)
[80-226] 164
(-2)
[100-235] 175
(-2)
[120-235] 186
(-2)
[150-225]
4 106
(+6)
[40-168] 113
(+4)
[40-178] 120
(+1)
[50-188] 128
(0)
[60-197] 136
(-2)
[70-206] 145
(-3)
[80-215] 155
(-3)
[90-224] 165
(-3)
[110-224] 175
(-2)
[130-214]
3 98
(+9)
[30-148] 102
(+5)
[30-158] 108
(+2)
[40-168] 114
(-1)
[50-177] 121
(-3)
[60-186] 129
(-5)
[70-195] 137
(-6)
[80-204] 145
(-7)
[90-213] 154
(-6)
[100-213] 164
(-2)
[120-203]
2 91
(+11)
[20-128] 93
(+5)
[20-138] 97
(+1)
[30-148] 102
(-3)
[40-157] 108
(-6)
[50-166] 114
(-9)
[60-175] 121
(-12)
[70-184] 128
(-14)
[80-193] 136
(-16)
[90-202] 144
(-18)
[100-202]
1 82
(+12)
[10-109] 86
(+9)
[10-119] 91
(+6)
[20-129] 97
(+4)
[30-138] 103
(+2)
[40-147] 110
(0)
[50-156] 117
(-1)
[60-165] 124
(-3)
[70-174] 132
(-4)
[80-183] 140
(-5)
[90-192] 149
(-4)
[100-192]
0 74
(+14)
[0-100] 80
(+12)
[10-109] 86
(+11)
[20-118] 92
(+9)
[30-127] 98
(+7)
[40-136] 105
(+6)
[50-145] 112
(+4)
[60-154] 120
(+4)
[70-163] 128
(+2)
[80-172] 136
(+1)
[90-181] 146
(+1)
[101-190]

	Number of spares
Number of strikes	0	1	2	3	4	5	6	7	8	9	10
12	300 (0) [300-300]
11	258 (-1) [240-299]	277 (0) [270-290]
10	223 (0) [180-288]	239 (-1) [210-289]	257 (0) [240-280]
9	193 (0) [120-268]	208 (0) [150-278]	223 (-1) [180-279]	239 (0) [210-269]
8	168 (0) [90-248]	181 (0) [110-258]	195 (-1) [130-268]	209 (-1) [160-268]	224 (0) [190-258]
7	147 (+1) [70-228]	159 (0) [80-238]	171 (0) [100-248]	184 (0) [120-257]	197 (-1) [150-257]	210 (-1) [180-247]
6	131 (+3) [60-208]	141 (+2) [60-218]	151 (0) [70-228]	162 (-1) [90-237]	173 (-2) [110-246]	185 (-2) [130-246]	198 (-1) [160-236]
5	117 (+5) [50-188]	125 (+2) [50-198]	134 (+1) [60-208]	143 (-1) [70-217]	153 (-2) [80-226]	164 (-2) [100-235]	175 (-2) [120-235]	186 (-2) [150-225]
4	106 (+6) [40-168]	113 (+4) [40-178]	120 (+1) [50-188]	128 (0) [60-197]	136 (-2) [70-206]	145 (-3) [80-215]	155 (-3) [90-224]	165 (-3) [110-224]	175 (-2) [130-214]
3	98 (+9) [30-148]	102 (+5) [30-158]	108 (+2) [40-168]	114 (-1) [50-177]	121 (-3) [60-186]	129 (-5) [70-195]	137 (-6) [80-204]	145 (-7) [90-213]	154 (-6) [100-213]	164 (-2) [120-203]
2	91 (+11) [20-128]	93 (+5) [20-138]	97 (+1) [30-148]	102 (-3) [40-157]	108 (-6) [50-166]	114 (-9) [60-175]	121 (-12) [70-184]	128 (-14) [80-193]	136 (-16) [90-202]	144 (-18) [100-202]
1	82 (+12) [10-109]	86 (+9) [10-119]	91 (+6) [20-129]	97 (+4) [30-138]	103 (+2) [40-147]	110 (0) [50-156]	117 (-1) [60-165]	124 (-3) [70-174]	132 (-4) [80-183]	140 (-5) [90-192]	149 (-4) [100-192]
0	74 (+14) [0-100]	80 (+12) [10-109]	86 (+11) [20-118]	92 (+9) [30-127]	98 (+7) [40-136]	105 (+6) [50-145]	112 (+4) [60-154]	120 (+4) [70-163]	128 (+2) [80-172]	136 (+1) [90-181]	146 (+1) [101-190]

As in ten-pin bowling, there are 438,578,270,526 clean games in which all 10 frames are closed, since the third ball has no effect on the number of closed frames.

Derivation of the Total Number of Possible Games

There are 286 possible ways to throw one, two or three balls in each of frames 1-9. To see this, let 0 represent a gutter ball and 10 a strike. The possiblities are shown in the following table:

First ball
A Second ball
B Third ball
C Possibilities Subtotal

Strike
10 (not thrown) (not thrown) 1 1

9 0 0 or 1 2 3

Spare
1 (not thrown) 1

8 0 0 to 2 3 6

1 0 or 1 2

Spare
2 (not thrown) 1

7 0 0 to 3 4 10

1 0 to 2 3

2 0 or 1 2

Spare
3 (not thrown) 1

...

0 0 0 to 10 11 66

1 0 to 9 10

...

9 0 or 1 2

Spare
10 (not thrown) 1

Total 286

First ball A	Second ball B	Third ball C	Possibilities	Subtotal
Strike 10	(not thrown)	(not thrown)	1	1
9	0	0 or 1	2	3
Spare 1	(not thrown)	1
8	0	0 to 2	3	6
1	0 or 1	2
Spare 2	(not thrown)	1
7	0	0 to 3	4	10
1	0 to 2	3
2	0 or 1	2
Spare 3	(not thrown)	1
...
0	0	0 to 10	11	66
1	0 to 9	10
...
9	0 or 1	2
Spare 10	(not thrown)	1
Total	286

In mathematical terms, the number of possibilities for one frame can be written as

Since any of the 286 possibilities for the first frame can be followed by any of them for the second frame, and so on, it follows that the number of possible ways to play the first nine frames is 286⁹.

There are more possibilities in the tenth frame due to the bonus balls:

First ball
A Second ball
B Third ball
C Possibilities Number of
strikes Number of
spares

Strike
10 Strike
10 Strike
10 1 3 0

Non-strike
0 to 9 10 2 0

Non-strike
0 to 9 Spare
10-B 10 1 1

Non-spare
0 to 9-B 55* 1 0

Non-strike
0 to 9 Spare
10-A Strike
10 10 1 1

Non-strike
0 to 9 100 0 1

Non-spare
0 to 9-A Open
0 to 10-A-B 275** 0 0

Total 461

* As B varies from 0 to 9, the sum of the 10-B possibilities 0 to 9-B is:

** The possibilities are:

First ball A	Second ball B	Third ball C	Possibilities	Number of strikes	Number of spares
Strike 10	Strike 10	Strike 10	1	3	0
Non-strike 0 to 9	10	2	0
Non-strike 0 to 9	Spare 10-B	10	1	1
Non-spare 0 to 9-B	55*	1	0
Non-strike 0 to 9	Spare 10-A	Strike 10	10	1	1
Non-strike 0 to 9	100	0	1
Non-spare 0 to 9-A	Open 0 to 10-A-B	275**	0	0
Total	461

Since there are 461 possible ways to score the tenth frame, the result given at the top of this section for the total number of games follows.

Candlepin/Duckpin Score Distribution Table

Here's a full table showing the number of possible games for each score in candlepin or duckpin bowling:

Score Number of possible games

0 1

1 30

2 465

3 4960

4 40920

5 278256

6 1623160

7 8347680

8 38608020

9 163011640

10 635745396

11 2311800564

12 7898630488

13 25518377930

14 78375432192

15 229884291082

16 646451520650

17 1748732389740

18 4563966767050

19 11521332965360

20 28195300168536

21 67022483942610

22 155020952858610

23 349427328192852

24 768628298733384

25 1651966406406522

26 3472864928181523

27 7148323121225494

28 14418997854579253

29 28525151936663804

30 55385606549260712

31 105615202604948957

32 197913164668393077

33 364650010275350069

34 660913262213665103

35 1178891078820558207

36 2070336182395359030

37 3581024808259975717

38 6102671860803001556

39 10249738663868045596

40 16971001012538287584

41 27708608953281893195

42 44620559750271008778

43 70885795636729790802

44 111114140570787462861

45 171885416888868395878

46 262442515545387531149

47 395560879117283161706

48 588610537683116599598

49 864815138783419166142

50 1254693995960731102537

51 1797646980476968342279

52 2543607620468429055057

53 3554647542239861745988

54 4906367257711503634629

55 6688857890478569714101

56 9006971369402871029155

57 11979600573276136579429

58 15737655290927971430631

59 20420435074909224118183

60 26170156283204797059848

61 33124496033363042570738

62 41407174596093172213215

63 51116807812344559744945

64 62314512039748314335577

65 75011015957016248688076

66 89154296623525046688698

67 104618973683468061217445

68 121198822999390603283101

69 138603767531028548448834

70 156462535013152540694779

71 174331820739543560595556

72 191712264491635662808520

73 208070876741419095525039

74 222868793939561511065001

75 235592495419431313947969

76 245785981776200464270566

77 253081005925114044395742

78 257222357592307499334918

79 258085489212664863100540

80 255684443534144785717602

81 250169044994429779565231

82 241811529062189273628191

83 230984036929714064043964

84 218129504604451418012410

85 203729244560827402543341

86 188270818920289689358561

87 172219571032916229160218

88 155996443369219089550939

89 139963584855536353903400

90 124417944639962013338573

91 109591814434343862275838

92 95658366850399709140548

93 82739820333054653646941

94 70915992856405721834427

95 60231586605998378164890

96 50701373040939563083457

97 42313310139045150012090

98 35030347089788099707772

99 28792090403446216702436

100 23517516275755793313709

101 19109497754383725029438

102 15461185213933660596255

103 12463464920695184620652

104 10012149212282694139132

105 8013504895878972813898

106 6387278262523817263927

107 5067215006895278299411

108 3999696824989735545934

109 3141323178206713139221

110 2456282292477494349356

111 1914173806652190955729

112 1488628601534690534006

113 1156708472915228014238

114 898796775179566257800

115 698605101264116959958

116 543043772290870082948

117 421899587994816038465

118 327382490575979012391

119 253615427739236267683

120 196142995020034189803

121 151518728092350513755

122 117005588526426529592

123 90390011471395193189

124 69881799065766862043

125 54056633824373828874

126 41807724302811120785

127 32293880027759478321

128 24886646513386714471

129 19119519104756754309

130 14642903951918636993

131 11187660900873748473

132 8539216431233998932

133 6521911309818926355

134 4991364651717932942

135 3830819543529840778

136 2948382354434072170

137 2273813255421464573

138 1754939833121898989

139 1353841660075053015

140 1043159847122700960

141 802839819938030316

142 617637816260865282

143 475484449774261096

144 366633474638098050

145 283254961953810767

146 219185877964053862

147 169687636815771259

148 131210807422085454

149 101157628459778481

150 77654423438096051

151 59337793519838010

152 45176362079801134

153 34339496382271391

154 26126586159034671

155 19942152363507767

156 15292971286073798

157 11783927087793304

158 9111757530154630

159 7054073191970686

160 5455028685159556

161 4207814162584108

162 3237755891380477

163 2488376524582342

164 1913647181811822

165 1475006581167232

166 1140669438081146

167 884967616203672

168 687841815126895

169 534189572470797

170 413203601767024

171 317472572763179

172 242022522405026

173 183278522630477

174 138301253566063

175 104407778538213

176 79189180844522

177 60534483379146

178 46681580010945

179 36238893993340

180 28197584774439

181 21887665069975

182 16903008379242

183 12987149908756

184 9947909051893

185 7617109331594

186 5848701739783

187 4515515266808

188 3510103638537

189 2743516965650

190 2147183589728

191 1672097746566

192 1288259812670

193 979276673597

194 735738141264

195 548354730133

196 407981489222

197 305379544392

198 231746331186

199 178901567625

200 140054116561

201 110125920523

202 86162686450

203 66762831555

204 51266159752

205 39076923625

206 29677756545

207 22572085831

208 17313342634

209 13462724719

210 10626431928

211 8449283427

212 6679432528

213 5180683382

214 3936713799

215 2929315798

216 2142226380

217 1551092713

218 1129585428

219 842289882

220 652314635

221 520331549

222 418100847

223 331844762

224 259822078

225 200213610

226 151984517

227 113751043

228 84691419

229 63519585

230 49078877

231 39437535

232 32563940

233 26339327

234 20796077

235 15908213

236 11765472

237 8359780

238 5760838

239 3920658

240 2802620

241 2192849

242 1837414

243 1521993

244 1245812

245 999254

246 785329

247 597204

248 439635

249 310271

250 216899

251 159604

252 140221

253 120812

254 102282

255 83568

256 66036

257 48991

258 34032

259 21022

260 12046

261 7441

262 6629

263 5837

264 5140

265 4431

266 3795

267 3126

268 2502

269 1825

270 1176

271 497

272 477

273 443

274 416

275 373

276 335

277 279

278 226

279 153

280 81

281 15

282 15

283 14

284 14

285 13

286 13

287 12

288 12

289 11

290 11

291 1

292 1

293 1

294 1

295 1

296 1

297 1

298 1

299 1

300 1

Total 5726805883325784576

Score	Number of possible games
0	1
1	30
2	465
3	4960
4	40920
5	278256
6	1623160
7	8347680
8	38608020
9	163011640
10	635745396
11	2311800564
12	7898630488
13	25518377930
14	78375432192
15	229884291082
16	646451520650
17	1748732389740
18	4563966767050
19	11521332965360
20	28195300168536
21	67022483942610
22	155020952858610
23	349427328192852
24	768628298733384
25	1651966406406522
26	3472864928181523
27	7148323121225494
28	14418997854579253
29	28525151936663804
30	55385606549260712
31	105615202604948957
32	197913164668393077
33	364650010275350069
34	660913262213665103
35	1178891078820558207
36	2070336182395359030
37	3581024808259975717
38	6102671860803001556
39	10249738663868045596
40	16971001012538287584
41	27708608953281893195
42	44620559750271008778
43	70885795636729790802
44	111114140570787462861
45	171885416888868395878
46	262442515545387531149
47	395560879117283161706
48	588610537683116599598
49	864815138783419166142
50	1254693995960731102537
51	1797646980476968342279
52	2543607620468429055057
53	3554647542239861745988
54	4906367257711503634629
55	6688857890478569714101
56	9006971369402871029155
57	11979600573276136579429
58	15737655290927971430631
59	20420435074909224118183
60	26170156283204797059848
61	33124496033363042570738
62	41407174596093172213215
63	51116807812344559744945
64	62314512039748314335577
65	75011015957016248688076
66	89154296623525046688698
67	104618973683468061217445
68	121198822999390603283101
69	138603767531028548448834
70	156462535013152540694779
71	174331820739543560595556
72	191712264491635662808520
73	208070876741419095525039
74	222868793939561511065001
75	235592495419431313947969
76	245785981776200464270566
77	253081005925114044395742
78	257222357592307499334918
79	258085489212664863100540
80	255684443534144785717602
81	250169044994429779565231
82	241811529062189273628191
83	230984036929714064043964
84	218129504604451418012410
85	203729244560827402543341
86	188270818920289689358561
87	172219571032916229160218
88	155996443369219089550939
89	139963584855536353903400
90	124417944639962013338573
91	109591814434343862275838
92	95658366850399709140548
93	82739820333054653646941
94	70915992856405721834427
95	60231586605998378164890
96	50701373040939563083457
97	42313310139045150012090
98	35030347089788099707772
99	28792090403446216702436
100	23517516275755793313709
101	19109497754383725029438
102	15461185213933660596255
103	12463464920695184620652
104	10012149212282694139132
105	8013504895878972813898
106	6387278262523817263927
107	5067215006895278299411
108	3999696824989735545934
109	3141323178206713139221
110	2456282292477494349356
111	1914173806652190955729
112	1488628601534690534006
113	1156708472915228014238
114	898796775179566257800
115	698605101264116959958
116	543043772290870082948
117	421899587994816038465
118	327382490575979012391
119	253615427739236267683
120	196142995020034189803
121	151518728092350513755
122	117005588526426529592
123	90390011471395193189
124	69881799065766862043
125	54056633824373828874
126	41807724302811120785
127	32293880027759478321
128	24886646513386714471
129	19119519104756754309
130	14642903951918636993
131	11187660900873748473
132	8539216431233998932
133	6521911309818926355
134	4991364651717932942
135	3830819543529840778
136	2948382354434072170
137	2273813255421464573
138	1754939833121898989
139	1353841660075053015
140	1043159847122700960
141	802839819938030316
142	617637816260865282
143	475484449774261096
144	366633474638098050
145	283254961953810767
146	219185877964053862
147	169687636815771259
148	131210807422085454
149	101157628459778481
150	77654423438096051
151	59337793519838010
152	45176362079801134
153	34339496382271391
154	26126586159034671
155	19942152363507767
156	15292971286073798
157	11783927087793304
158	9111757530154630
159	7054073191970686
160	5455028685159556
161	4207814162584108
162	3237755891380477
163	2488376524582342
164	1913647181811822
165	1475006581167232
166	1140669438081146
167	884967616203672
168	687841815126895
169	534189572470797
170	413203601767024
171	317472572763179
172	242022522405026
173	183278522630477
174	138301253566063
175	104407778538213
176	79189180844522
177	60534483379146
178	46681580010945
179	36238893993340
180	28197584774439
181	21887665069975
182	16903008379242
183	12987149908756
184	9947909051893
185	7617109331594
186	5848701739783
187	4515515266808
188	3510103638537
189	2743516965650
190	2147183589728
191	1672097746566
192	1288259812670
193	979276673597
194	735738141264
195	548354730133
196	407981489222
197	305379544392
198	231746331186
199	178901567625
200	140054116561
201	110125920523
202	86162686450
203	66762831555
204	51266159752
205	39076923625
206	29677756545
207	22572085831
208	17313342634
209	13462724719
210	10626431928
211	8449283427
212	6679432528
213	5180683382
214	3936713799
215	2929315798
216	2142226380
217	1551092713
218	1129585428
219	842289882
220	652314635
221	520331549
222	418100847
223	331844762
224	259822078
225	200213610
226	151984517
227	113751043
228	84691419
229	63519585
230	49078877
231	39437535
232	32563940
233	26339327
234	20796077
235	15908213
236	11765472
237	8359780
238	5760838
239	3920658
240	2802620
241	2192849
242	1837414
243	1521993
244	1245812
245	999254
246	785329
247	597204
248	439635
249	310271
250	216899
251	159604
252	140221
253	120812
254	102282
255	83568
256	66036
257	48991
258	34032
259	21022
260	12046
261	7441
262	6629
263	5837
264	5140
265	4431
266	3795
267	3126
268	2502
269	1825
270	1176
271	497
272	477
273	443
274	416
275	373
276	335
277	279
278	226
279	153
280	81
281	15
282	15
283	14
284	14
285	13
286	13
287	12
288	12
289	11
290	11
291	1
292	1
293	1
294	1
295	1
296	1
297	1
298	1
299	1
300	1
Total	5726805883325784576

The total number of games is more than 2⁶³ but less than 2⁹⁵, so 12-byte signed integer arithmetic can be used for exact counts of the numbers of games with each score.

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