Saturday, March 7, 2009

Which pin is the most likely to fall?

After one toss of the bowling ball, which pin or pins are the most likely to fall? Think about this problem for a minute because I am soon going to give you the answer. There are some obvious answers. Since the most common target for a right-handed bowler is the 1-3 pocket, one might expect it to be one of these pins. For a left-handed bowler, it would be the 1-2 pocket. None of these pins is the correct answer. Could it be the 4 or the 6 pin? If you hit or miss the pocket, these pins would be the next most likely to go down. It is neither of these. If you guessed the 5 pin, you are also incorrect. The most likely pin(s) to fall are in the back row. The answer? It is the 8 pin. Before we explore why that is the case, first I'll tell you how I answered the question.

For approximately the last four weeks, I have been recording spare combinations with the intent of determining how many possible spares exist in bowling. I'm not quite ready to answer that question yet, but over this time, I have recorded leaves, or the pins standing after the first ball is thrown. Of these, many are repeated. In fact, only 1 in 4 (approximately) is unique in my current sample. Nonetheless, using these data, it is possible to see how often each of the pins remains standing or goes down after the first roll. I must admit that I do not know how universal these results are, meaning that I don't know if they would apply to all bowlers. They were primarily accumulated for our team and opponents, which range in skill from ca. 128 to 200+ average bowlers. Approximately 90% of these were left by right-handed bowlers, and 10% by south paws.

The graph above shows the frequency with which each pin has been knocked down over the observation period. There are a few general patterns that can be identified. 1-The likelihood of a pin falling increases by row. The back row pins are the most likely to go down, and the front row (1 pin) is the least likely. 2-The 1 and 5 pins are the most likely to remain standing after the first ball, and were not knocked down in approximately 2 of 3 attempts. 3-For the 3rd row (4, 5, & 6 pins), the outer pins are more likely to fall than the central 5 pin. For the back row, the opposite is true. The central pins (8 & 9) are more likely to fall than the corner pins (7 & 10). 4-For every symmetrical pair of pins (2 vs. 3, 4 vs. 6, 7 vs. 10, 8 vs. 9), the pin on the left side of the rack is more likely to fall than the one on the right, something which is likely attributable to right-handedness.

In the figure above, I have removed the asymmetry due to handedness and shown the same data as a bubble graph. Each bubble represents the pin in its position. Larger bubbles mean a greater likelihood of the pin going down after the 1st ball is thrown. The reason why the pins of the back row are the most likely to fall is simple. Except for a ball thrown in the gutter, a back row pin will always go down, or almost always (see here). If the 1 pin is not struck with the ball, it will rarely go down. I can't say that I understand the hardiness of the 5 pin as it should commonly be in the thick of the action, but it remains standing as often as the 1 pin.

So why is it that the 8 and 9 pins are the most likely to fall? Although I'm not sure, the answer must have to do with pin action. In essence, many paths lead to the 8 and 9. A ball thrown straight down the middle will take out the 8 and 9. A ball that hits the 4 or 6 pin will almost always take out at least one. In fact, it's hard to imagine many ball strikes in which both the 8 and 9 are left untouched.

Of the 456 leaves I have recorded, only 24 or 5% have both the 8 and 9 standing after the 1st ball, and of these 15 involve a gutter ball or only the 7 or 10 pin picked up. The remaining nine are all unique, and every single one couples the 8 and 9 pin with the 1 and/or 5 pin. In short, in order to leave the 8 and 9 standing requires throwing a ball off center and getting some unusual pin action. Otherwise, if you pick up at least two pins, at least one is very likely to go down, somewhere in the neighborhood of a 98% chance.


  1. Awesome! Good work! I guessed the 5 pin, and boy was I wrong!

  2. Thanks Ring, I would have never guessed the 8 either. It is very counter intuitive and yet it also make sense.


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