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ALL ABOUT BOWLING SCORES |  |
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ALL ABOUT BOWLING SCORES | |
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 11¹º = 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 is
66^9 x 241 = 5,726,805,883,325,784,576 (about 6 billion billion, or 6 quintillion)Calculations of the score distributions for these games results in the following table for the number of possible games and the associated probability if all scores were 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 %
...
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
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!
If we show the cumulative probability histogram corresponding to the score distribution, we can get an idea of 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.
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 score contour lines in this diagram how much we can expect to increase our average score with more 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 |
| 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] |
| 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] |
| 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] |
|
| 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] |
|
| 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] |
||
| 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] |
|||
| 6 | 128 [60-204] |
139 [70-215] |
151 [80-226] |
163 [100-236] |
175 [120-246] |
187 [140-246] |
199 [170-235] |
||||
| 7 | 146 [70-225] |
159 [90-236] |
171 [110-247] |
184 [130-257] |
198 [150-257] |
211 [180-246] |
|||||
| 8 | 168 [90-246] |
181 [120-257] |
196 [140-268] |
210 [170-268] |
224 [200-257] |
||||||
| 9 | 193 [120-267] |
208 [150-278] |
224 [180-279] |
239 [210-268] |
|||||||
| 10 | 223 [180-288] |
240 [210-289] |
257 [240-279] |
||||||||
| 11 | 259 [240-299] |
277 [270-290] |
|||||||||
| 12 | 300 [300-300] |
||||||||||
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.
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 175. Obviously the only game with 12 strikes 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 90, 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. If A and B were the balls thrown in frame 10 (0 ≤ A ≤ 9, 0 ≤ A + B ≤ 9), then 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 + 2BThe 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 relatively-unusual 11-ball games is not apparent on the chart above.
Derivation of the total number of possible games. In each of frames 1 -
9, there are 66 possible ways to score the two balls thrown (or single ball in
the case of a strike). To see this, let 0 represent a gutter ball and 10 a
strike. The possibilities for the first ball A are in the range 0 to 10,
inclusive (0 ≤ A ≤ 10). If A is not a strike (A ≠ 10), then there are
(11 - A) possibilities for the second ball B:
0 ≤ B ≤ 10 - A. Including the strike possibility, the total number of
ball combinations for one frame then is
9 10-A 9 11
1 + ∑ ∑ 1 = 1 + ∑ (11-A) = 1 + ∑ A = 1 + 66 - 1 = 66
A=0 B=0 A=0 A=2
It follows that the number of possible ways to score the first nine frames is
66^9 (66 raised to the ninth power). 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 | ||||
9 9-B 9 9-A * ∑ ∑ 1 = 55 ** ∑ ∑ 1 = 55 B=0 C=0 A=0 B=0Since 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.
The "Whale" chart. If we plot the average, minimum and maximum score against the number of strikes and then repeat the plot for various numbers of spares, we get the following unusual diagram:
This diagram is vaguely reminiscent of the head and mouth of a humpback whale, but the information it contains is probably easier to derive from the score contour line diagram.
Score distribution table. Here's a full table showing the number of possible games for each 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
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 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.
Copyright © 2007 Balmoral Software. All rights reserved. Republication or redistribution is expressly prohibited without prior written consent.
