ALL ABOUT BOWLING SCORES

Ever wonder what the most common 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 ten-pin 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.

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 + 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 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=0
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.

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:

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 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.

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