Saturday, January 24, 2009

Bowling All Spares

In this post, I explore the total range of scores that can be received if a bowler receives a spare on every frame. When a spare is received, the pin total of the next toss is added to the frame with the spare. The answer depends upon the values of the first balls thrown, and there are a huge number of possible games. I’ll begin with the minimum and maximum.

The minimum bowling score possible in a game with 10 spares is 100. This is achieved by throwing gutter ball-spare on every frame with a gutter ball thrown for the 3rd ball in the 10th. In this game, a score of 10 would occur in every frame. The maximum possible score depends on whether you allow a strike for the 3rd ball in the 10th. I’ll assume that the score of the bonus ball is a nine. In this case, the maximum is 190. This is achieved by throwing 9-spare on every frame with 9-spare-9 in the 10th. This would equate to 10 consecutive frame scores of 19. So, a game comprised of nothing but spares can produced a score ranging from 100 to 190. If a strike is allowed for the bonus ball in the 10th, the maximum score is 191.

Neither of these outcomes is particularly likely because they would require a degree of consistency that is probably not attainable. For the minimum, it is hard to imagine how someone could consistently alternate between gutter balls and knocking down all ten pins. The maximum score is perhaps a bit more likely, but it makes you wonder how someone could consistently pick up single pin spares but not record a strike. So what are the most likely game scores in games comprised of all spares?

To answer this question, I performed two simple simulations. In each, I simulated 5,000 games. In the first simulation, a random number of pins (0 to 9) was given for the 1st ball in all frames and the 3rd ball in the 10th. The results are shown in the histogram above. The average score comes in at 145, and 95% of the scores would range between 127 and 163. This simulation probably is not a very good reflection of reality because the likelihood of picking up spares correlates with the 1st ball pin score (see here). For example, it is much easier to pick up a single pin spare than one involving seven pins left standing after the first throw.

In the second simulation, I changed only the range of the random numbers plugged into the 1st ball throws to vary between 6 and 9. In this version, the average score is 175, and 95% of scores fall between 168 and 182. This is probably closer to reality for the typical scores received in actual games involving nothing but spares.

1 comment:

  1. I think you'll be pleased to know that I am leaving your site happy. So happy, in fact, that I need a new keyboard.


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