Saturday, February 7, 2009

Chances of Bowling a Perfect Game

Have you ever wondered what are your chances of bowling a perfect game? A perfect game consists of 12 consecutive strikes totaling to a score of 300. It is actually a remarkably simple thing to estimate. For the Bowl Movements, a perfect game is nearly impossible at our current skill level. Since we have seen this happen at the Lanes at least three times, though, for "non-developing" bowlers it is not that uncommon.

First, I'll give you the calculation. I'll explain it later if you really want to know how its derived. In order to know your chances of rolling 12 straight strikes, you only have to know one thing, your strike probability. In a single game, you have between 10 and 12 opportunities for strikes depending upon what happens in the 10th frame. To calculate your strike probability, take the number of strikes you receive divided by the number of opportunities you have to bowl a strike. The larger your sample, the better. To get a really good idea of your strike probability, I recommend you have at least 30-50 frames. Once you have this number, raise it to the 12th power. This is the probability that you will roll 12 strikes in a row. You can multiply it by 100 if you want to convert it to a percentage, i.e., a 25% chance.The graph above shows the relationship between strike probability and perfect game probability. In brief, unless you strike >50% of your frames, perfect games will be extremely rare, generally occurring less than once in a thousand games. This aptly describes our game. My strike probability is 0.2368, or approximately one strike per 3.8 frames. This corresponds to a perfect game probability of 0.000000113, or one perfect game in every 9 million attempts. In comparison, PBA Bowler Walter Ray Williams had a strike percentage of 66.35% for the 2007-08 season (the best ever in PBA competition). This corresponds to a perfect game probability of 0.0072 (0.72%), or one in every 137 games.

The calculation is a simple case of binomial probability. Think of it like tossing a coin. When you toss a coin, the probability of getting heads is 50%. The probability of getting heads followed by another heads is .5 x .5 = 0.25. The probability of getting three heads in a row is .5 x .5 x .5, or 12.5%. The probability of getting 12 heads in a row is .5 to the 12th power, or 0.024%. It's very unlikely. (Go ahead and try it if you don't believe me, and you have a lot of time.) Calculating the likelihood of a perfect game is identical. Just replace heads and tails with strike and non-strike. If you know the probability of a strike, the probability of 12 in a row is that probability raised to the 12th.

I do wonder if another factor must be considered. I think bowlers, like any other sports folk are prone to hot streaks. If you find yourself "in the zone," and your motion is consistent, you may be able to temporarily elevate your strike probability. My hunch is that this is what perfect games are all about. So, this calculation may be somewhat of an underestimate.

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