Saturday, February 21, 2009

How many spare combinations are possible in bowling? Part I

In this post, I explore the issue of the number of possible spares that exist in bowling. What I mean is how many different combinations of pins can remain after the first ball is thrown. I do not explore whether those spares can actually be picked up. My suspicion is that there are certain spares that are impossible to convert, but this would be very difficult to establish.

[A note: To avoid confusion, I refer to individual pins using numerals. For example “2 pin” refers specifically to the 2 pin. I use words to refer to the number of pins remaining. For example, a “two pin spare” means that two pins remain after the first throw.]

There are two ways to address this question. First obviously there are a finite number of pin combinations that exist. For example, there is only one ten pin spare combination (i.e., the first ball was thrown in the gutter.) There are ten single pin spare possibilities. Any single pin can remain standing after one toss of the ball. Likewise, there are ten nine pin spare possibilities, if any single pin is removed by the first throw. It should be obvious, however, that while some of these spare combinations can exist in theory, in reality they cannot, at least in ten pin bowling. [After playing candlepins last December, it seemed like anything was possible in that crazy game.] For example, it is impossible to pick up only the 5 pin with the first toss. In fact, most nine pin spare combinations are not possible. My hunch is that if a single pin is picked up on the first throw, it will be either the 7 or 10. The remaining eight nine pin spare combinations are likely impossible.

I begin with the total number of pin combinations that can exist in theory. I then turn to the more difficult question of how many of these can actually occur in a game of bowling, but I leave the ultimate solution to that problem for another day [that day has finally arrived]. The first problem is fairly easy to solve. It turns out that there are 1,023 different combinations of pins that can theoretically remain after the first ball is thrown. The table and histogram below show how many possible spares there are given the number of pins remaining after the first throw. Notice that the distribution is nearly normal and symmetrical with one exception. There are no zero pin spare possibilities, but there is one ten pin spare. The greatest number of possible spare combinations is when five pins remain. Of these, there are 252.

Now for the hard part. How many of these can actually exist in the real world? The short answer is, “I don’t know.” Some spares are easy to eliminate. For example, I can’t imagine how a bowler could be left with a nine pin spare opportunity with only the 9 pin missing, or an eight pin spare in which the 7 and 10 have been removed. Others are more difficult. For example, is it possible to leave only the 1, 7, and 10 pins standing? It seems like it might be, but I am fairly certain that I have never seen it (but see here).

So, how should one go about solving this problem? I see two solutions, one empirical and one theoretical. Beginning with the latter, I suppose it would be possible to model or simulate bowling and pin action. If one could do this in a realistic way, it would be possible to simulate literally millions of possible throws and pin strikes to determine the entire range of possible outcomes. My hunch is that it that this system is so complex that it would be difficult to attain this level of realism, yes, even with Nintendo Wii.

The other option, though perhaps much more onerous is to actually record spares observed in the real world. As the sample gets larger, the number of observed spare combinations should asymptotically approach the actual number that can exist, meaning that the number of unique spares observed will begin to plateau. Using statistical methods, it should be possible to estimate with some degree of confidence the total number of combinations that can occur. To get a good estimate, ideally one would observe bowlers of low skill because I suspect that high skill bowlers see relatively few spare combinations compared to those whose first balls are much more erratic.

I have begun collecting this type of data, and I will report on the results in the future. For now, I will share that I have recorded 226 spare combos, and of these 85 are unique.

Here is a related post.
And here's another.
Here's Part II.


  1. this is a great blog. i am an avid fan of sports statistics, regardless of sport, because i believe it helps wonders in improving game IQ and setting strategy. keep up with the posts!

  2. Thanks Jonathan! You made my day. I couldn't agree more.


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