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 ofstrikes 0 1 2 3 4 5 6 7 8 300[300-300] 259[240-299] 277[270-290] 223[180-288] 240[210-289] 257[240-279] 193[120-267] 208[150-278] 224[180-279] 239[210-268] 168[90-246] 181[120-257] 196[140-268] 210[170-268] 224[200-257] 146[70-225] 159[90-236] 171[110-247] 184[130-257] 198[150-257] 211[180-246] 128[60-204] 139[70-215] 151[80-226] 163[100-236] 175[120-246] 187[140-246] 199[170-235] 112[50-183] 123[60-194] 133[70-205] 144[80-215] 155[90-225] 166[110-235] 177[130-235] 188[150-224] 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] 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] 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] 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] 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
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
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:
ScoreNumber of possible games
01
120
2210
31540
48855
542504
6177100
7657800
82220075
96906900
1020030010
1154627084
12141116637
13347336412
14818558424
151854631380
164053948342
178574134256
1817590903116
1935084425512
2068153183370
21129156542039
22239128282128
23433093980298
24768175029950
251335679056261
262278764308864
273817721269708
286285424931278
2910176048813473
3016210652213304
3125423690787719
3239274771758064
3359789973730461
3489736657900900
35132834787033075
36194006223597572
37279661205716974
38398018151390200
39559449136091831
40776838931567572
411065940588576732
421445705502357343
431938561121705315
442570605432880903
453371684590465908
464375319099346208
475618445228564793
487140942201229333
498984922304030443
5011193770355829009
5113810930667765157
5216878453276117746
5320435326129713654
5424515635362932954
5529146610869639549
5634346628376654913
5740123251227815383
5846471404549689351
5953371780703441318
6060789577452586487
6168673668434334934
6276956298564663402
6385553384395717227
6494365480254213528
65103279445170253902
66112170812747354087
67120906827121834566
68129350064451661348
69137362512979745598
70144809940796620325
71151566341291631624
72157518221668013078
73162568486673578693
74166639683923175378
75169676402232105648
76171646676234883305
77172542309343731946
78172378125687965848
79171190226627438257
80169033430825208027
81165978103316094584
82162106654714921075
83157509948809043576
84152283892386077931
85146526364181517039
86140334651650668803
87133803399444707801
88127023103852577896
89120079021507938035
90113050455155943519
91106010240661754449
9299024411737621323
9392151904402003308
9485444345654857875
9578945863453573001
9672693023944120045
9766714881583314335
9861033240145235763
9955663091133973346
10050613244155051856
10145887089510794122
10241483436078768079
10337397371704961189
10433621048067136846
10530144388614623696
10626955619314626157
10724041709119775647
10821388640692533960
10918981680119465910
11016805547548715206
11114844654231857239
11213083276623221517
11311505812292077067
11410096971927616045
1158842020009154293
1167726929590817265
1176738528470417086
1185864552560171552
1195093653838062639
1204415377510495980
1213820097597373727
1223298981687014508
1232843905747206868
1242447444695948898
1252102793053565659
1261803790254604935
1271544848145184291
1281320992367181792
1291127775864826813
130961294388171457
131818085023387881
132695128788327698
133589753122859383
134499630252931260
135422696870992462
136357151976811922
137301400973036441
138254052574077937
139213889601295347
140179862464456172
141151065169242834
142126722015973414
143106169469752641
14488840622360686
14574252067274687
14661990415093876
14751701385089887
14843082666091665
14935870481552300
15029843343433392
15124808172866872
15220607116162379
15317101443169235
15414181008701762
15511747089496422
1569723545122578
1578040378083433
1586644452641044
1595486702080236
1604529003381568
1613736165201688
1623081105018158
1632539255963377
1642091793858275
1651721930513702
1661416734360140
1671164733232308
168957190045595
169785911852914
170645295369580
171529489941608
172434606120455
173356481490646
174292487050484
175239755303889
176196550315542
177160954253448
178131791387388
179107847709116
18088241591630
18172162948863
18259038079745
18348284335855
18439509743432
18532308399043
18626423428886
18721582203262
18817624621529
18914368737009
19011720626558
1919552812749
1927790240907
1936351933169
1945185250585
1954232118751
1963457204258
1972821392492
1982302090127
1991874802017
2001526313637
2011239515641
2021007719386
203818568928
204666193896
205542061609
206442072320
207360234562
208293886739
209239045260
210194337731
211157306293
212127325163
213102799565
21483194097
21567300605
21654691522
21744477808
21836317458
21929606794
22024117404
22119554213
22215820964
22312736481
22410258846
2258244157
2266659561
2275381526
2284385243
2293576841
2302930385
2312376760
2321924226
2331541327
2341231527
235975760
236777090
237617547
238498228
239404981
240335065
241275998
242226966
243183727
244148442
245117291
24693525
24773010
24857960
24945826
25037965
25131193
25226131
25321406
25417422
25513613
25610696
2577975
2586005
2594374
2603534
2613016
2622635
2632264
2641933
2651603
2661323
2671045
268810
269585
270406
271277
272258
273227
274206
275173
276150
277115
27890
27953
28026
28115
28215
28314
28414
28513
28613
28712
28812
28911
29011
2911
2921
2931
2941
2951
2961
2971
2981
2991
3001
Total5726805883325784576

#### 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
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 9 9 9 20 72 77 7.65 7.42 7.65 13.84 76.5 79.7 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 30T 10P 0 30T + 10P 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 ofstrikes 0 1 2 3 4 5 6 7 8 300(+77)[300-300] 276(+83)[270-279] 285(+77)[280-289] 252(+84)[240-258] 261(+80)[250-268] 269(+73)[260-278] 228(+82)[210-237] 237(+78)[220-247] 245(+74)[230-257] 254(+70)[240-267] 204(+76)[180-216] 213(+74)[190-226] 221(+70)[200-236] 230(+67)[210-246] 238(+63)[220-256] 180(+68)[150-195] 189(+66)[160-205] 197(+64)[170-215] 206(+62)[180-225] 214(+59)[190-235] 223(+57)[200-245] 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] 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] 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] 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] 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 75.0 73 74 75.8 72 75 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:
ScoreNumber of possible games
01
120
2210
31540
48855
542504
6177100
7657800
82220075
96906900
1020029910
1154625390
12141101325
13347238500
14818062300
151852514070
164046044310
178547655600
1817509826400
1934854380550
2067541888900
21127622461010
22235467859445
23424744086650
24749883201650
251297052375366
262199889956935
273661566327540
285985003339040
299613307642730
3015182582176769
3123589203727170
3236073308835395
3354319527255130
3480574860327825
35117781961461410
36169723917774345
37241173996359780
38338040825065020
39467496338531750
40638070844965110
41859697155357480
421143684325691265
431502601676134770
441950055813772110
452500347731015040
463168003824781765
473967183760495780
484910979046898705
496010628213027240
507274686502965781
518708198717340170
5210311931823519525
5312081727884792140
5414008036633963050
5516075680003223462
5618263888111000370
5720546628328775240
5822893227578599125
5925269264984684270
6027637689796314787
6129960100551745920
6232198107959174615
6334314697579840260
6436275509938490805
6538049964968674380
6639612173449501415
6740941598074127430
6842023448224326255
6942848812855238280
7043414552998372050
7143722987768421870
7243781414679391290
7343601506663391340
7443198625417862815
7542591085277769086
7641799395825565305
7740845506909141420
7839752077718176655
7938541791092813400
8037236734265954227
8135857866748978560
8234424594142676465
8332954462581014750
8431462981911416615
8529963576660029292
8628467653020803480
8726984759038298590
8825522806327282305
8924088318835376660
9022686676709427605
9121322330217135730
9219998968445645700
9318719638387834240
9417486820124802685
9516302471277776220
9615168057254583800
9714084582319246780
9813052630658586615
9912072418722804900
10011143854188694724
10110266593344594420
1029440087767430720
1038663612863528440
1047936274875808590
1057256998619048336
1066624504282096975
1076037286287205860
1085493607797089085
1094991517434455090
1104528888417358896
1114103475135395670
1123712978649550105
1133355111029457150
1143027649006567760
1152728470033340074
1162455568079499915
1172207051522130810
1181981129914883315
1191776098207280280
1201590324559115145
1211422245289812010
1221270367906237890
1231133280782366540
1241009666167227130
125898312049098324
126798118306584825
127708093859220590
128627344542961095
129555056550996240
130490479529459266
131432912437384680
132381694130477375
133336199390618620
134295839903993625
135260068624420566
136228385209907195
137200339988923560
138175534435457010
139153617687340420
140134280575449174
141117248552344640
142102274752682935
14389134183607230
14477619732640965
14567540296613840
14658720883516950
14751004028108040
14844251302626970
14938343110315140
15033177167222081
15128666108631730
15224734664812935
15321316832602560
15418353420191210
15515790257735412
15613577241099990
15711668204865040
15810021398119875
1598600056590950
1607372547220108
1616312149354020
1625396549489370
1634607137627000
1643928202327005
1653346128105804
1662848702430145
1672424639770410
1682063426478455
1691755582128400
1701492817304730
1711268092189410
1721075584180220
173910576986280
174769288201635
175648657261636
176546120947470
177459409229300
178386400235770
179325079514760
180273596522207
181230305907930
182193793062990
183162883715630
184136637715655
185114327530998
18695402395945
18779439498830
18866084078730
18954980816560
19045786537349
19138181246870
19231877185060
19326625569460
19422220692830
19518501030150
19615347001405
19712675027870
19810427511060
1998558355090
2007019752752
2015763798430
2024744025670
2033916846800
2043242871225
2052688078252
2062224819530
2071832625420
2081498788840
2091218699360
210987127782
211798400920
212646587315
213525693840
214429873150
215353641932
216292109910
217241219560
218197996490
219160810440
220129328302
221103144140
22281787560
22364732620
22451407280
22541203392
22633487230
22727610560
22822922250
22918780420
23015177312
23112097680
2329518700
2337409880
2345732970
2354441872
2363482550
2372792940
2382302860
2391933920
2401599447
2411300020
2421035990
243807480
244614385
245456372
246332880
247243120
248186075
249160500
250136530
251114270
25293825
25375300
25458800
25544430
25632295
25722500
25815150
25910350
2609225
2618130
2627065
2636030
2645025
2654050
2663105
2672190
2681305
269450
270415
271380
272345
273310
274275
275240
276205
277170
278135
279100
28010
28110
28210
28310
28410
28510
28610
28710
28810
28910
2900
2910
2920
2930
2940
2950
2960
2970
2980
2990
3001
Total1568336880910795776

# 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 ofstrikes 0 1 2 3 4 5 6 7 8 300(0)[300-300] 258(-1)[240-299] 277(0)[270-290] 223(0)[180-288] 239(-1)[210-289] 257(0)[240-280] 193(0)[120-268] 208(0)[150-278] 223(-1)[180-279] 239(0)[210-269] 168(0)[90-248] 181(0)[110-258] 195(-1)[130-268] 209(-1)[160-268] 224(0)[190-258] 147(+1)[70-228] 159(0)[80-238] 171(0)[100-248] 184(0)[120-257] 197(-1)[150-257] 210(-1)[180-247] 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] 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] 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] 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] 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] 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] 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
PossibilitiesSubtotal
Strike
10
(not thrown)(not thrown)11
900 or 123
Spare
1
(not thrown)1
800 to 236
10 or 12
Spare
2
(not thrown)1
700 to 3410
10 to 23
20 or 12
Spare
3
(not thrown)1
...
000 to 101166
10 to 910
...
90 or 12
Spare
10
(not thrown)1
Total286
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
PossibilitiesNumber of
strikes
Number of
spares
Strike
10
Strike
10
Strike
10
130
Non-strike
0 to 9
1020
Non-strike
0 to 9
Spare
10-B
1011
Non-spare
0 to 9-B
55*10
Non-strike
0 to 9
Spare
10-A
Strike
10
1011
Non-strike
0 to 9
10001
Non-spare
0 to 9-A
Open
0 to 10-A-B
275**00
Total461

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

** The possibilities are:
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:
ScoreNumber of possible games
01
130
2465
34960
440920
5278256
61623160
78347680
838608020
9163011640
10635745396
112311800564
127898630488
1325518377930
1478375432192
15229884291082
16646451520650
171748732389740
184563966767050
1911521332965360
2028195300168536
2167022483942610
22155020952858610
23349427328192852
24768628298733384
251651966406406522
263472864928181523
277148323121225494
2814418997854579253
2928525151936663804
3055385606549260712
31105615202604948957
32197913164668393077
33364650010275350069
34660913262213665103
351178891078820558207
362070336182395359030
373581024808259975717
386102671860803001556
3910249738663868045596
4016971001012538287584
4127708608953281893195
4244620559750271008778
4370885795636729790802
44111114140570787462861
45171885416888868395878
46262442515545387531149
47395560879117283161706
48588610537683116599598
49864815138783419166142
501254693995960731102537
511797646980476968342279
522543607620468429055057
533554647542239861745988
544906367257711503634629
556688857890478569714101
569006971369402871029155
5711979600573276136579429
5815737655290927971430631
5920420435074909224118183
6026170156283204797059848
6133124496033363042570738
6241407174596093172213215
6351116807812344559744945
6462314512039748314335577
6575011015957016248688076
6689154296623525046688698
67104618973683468061217445
68121198822999390603283101
69138603767531028548448834
70156462535013152540694779
71174331820739543560595556
72191712264491635662808520
73208070876741419095525039
74222868793939561511065001
75235592495419431313947969
76245785981776200464270566
77253081005925114044395742
78257222357592307499334918
79258085489212664863100540
80255684443534144785717602
81250169044994429779565231
82241811529062189273628191
83230984036929714064043964
84218129504604451418012410
85203729244560827402543341
86188270818920289689358561
87172219571032916229160218
88155996443369219089550939
89139963584855536353903400
90124417944639962013338573
91109591814434343862275838
9295658366850399709140548
9382739820333054653646941
9470915992856405721834427
9560231586605998378164890
9650701373040939563083457
9742313310139045150012090
9835030347089788099707772
9928792090403446216702436
10023517516275755793313709
10119109497754383725029438
10215461185213933660596255
10312463464920695184620652
10410012149212282694139132
1058013504895878972813898
1066387278262523817263927
1075067215006895278299411
1083999696824989735545934
1093141323178206713139221
1102456282292477494349356
1111914173806652190955729
1121488628601534690534006
1131156708472915228014238
114898796775179566257800
115698605101264116959958
116543043772290870082948
117421899587994816038465
118327382490575979012391
119253615427739236267683
120196142995020034189803
121151518728092350513755
122117005588526426529592
12390390011471395193189
12469881799065766862043
12554056633824373828874
12641807724302811120785
12732293880027759478321
12824886646513386714471
12919119519104756754309
13014642903951918636993
13111187660900873748473
1328539216431233998932
1336521911309818926355
1344991364651717932942
1353830819543529840778
1362948382354434072170
1372273813255421464573
1381754939833121898989
1391353841660075053015
1401043159847122700960
141802839819938030316
142617637816260865282
143475484449774261096
144366633474638098050
145283254961953810767
146219185877964053862
147169687636815771259
148131210807422085454
149101157628459778481
15077654423438096051
15159337793519838010
15245176362079801134
15334339496382271391
15426126586159034671
15519942152363507767
15615292971286073798
15711783927087793304
1589111757530154630
1597054073191970686
1605455028685159556
1614207814162584108
1623237755891380477
1632488376524582342
1641913647181811822
1651475006581167232
1661140669438081146
167884967616203672
168687841815126895
169534189572470797
170413203601767024
171317472572763179
172242022522405026
173183278522630477
174138301253566063
175104407778538213
17679189180844522
17760534483379146
17846681580010945
17936238893993340
18028197584774439
18121887665069975
18216903008379242
18312987149908756
1849947909051893
1857617109331594
1865848701739783
1874515515266808
1883510103638537
1892743516965650
1902147183589728
1911672097746566
1921288259812670
193979276673597
194735738141264
195548354730133
196407981489222
197305379544392
198231746331186
199178901567625
200140054116561
201110125920523
20286162686450
20366762831555
20451266159752
20539076923625
20629677756545
20722572085831
20817313342634
20913462724719
21010626431928
2118449283427
2126679432528
2135180683382
2143936713799
2152929315798
2162142226380
2171551092713
2181129585428
219842289882
220652314635
221520331549
222418100847
223331844762
224259822078
225200213610
226151984517
227113751043
22884691419
22963519585
23049078877
23139437535
23232563940
23326339327
23420796077
23515908213
23611765472
2378359780
2385760838
2393920658
2402802620
2412192849
2421837414
2431521993
2441245812
245999254
246785329
247597204
248439635
249310271
250216899
251159604
252140221
253120812
254102282
25583568
25666036
25748991
25834032
25921022
26012046
2617441
2626629
2635837
2645140
2654431
2663795
2673126
2682502
2691825
2701176
271497
272477
273443
274416
275373
276335
277279
278226
279153
28081
28115
28215
28314
28414
28513
28613
28712
28812
28911
29011
2911
2921
2931
2941
2951
2961
2971
2981
2991
3001
Total5726805883325784576

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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