## Bowling Simulation

I don’t have a particularly good question here.  I mainly just felt like writing a bowling simulation program and explaining the results.  If I had to form a question, I’d probably ask:  What does it take to become as good as a PBA bowler?

First off, I will ask the simple question of how likely is someone to roll a 300 game?  This is easy to answer even without a simulation program.  Rolling 300 involves rolling 12 consecutive strikes.  So, if someone rolls a strike 99% of the time, then their chances of rolling 300 are 99% raised to the 12th power, or about 88.6%.  Of course, no one is that good.  Based on my simulations and the fact that top PGA bowlers average around 220, I’d estimate that PBA bowlers throw strikes about 60% of the time.  At that rate, they should expect to roll a perfect 300 about once every 459 games.  (Note:  I just looked it up, and the record for season strike percentage is 66.35%, which would mean a perfect 300 about once every 137 games, a big difference — but 66.35 is the record, not the typical PBA bowler.)  I estimate my strike percentage as closer to 20%, meaning I’d roll a 300 about once every 244 million games.  Ha — not gonna happen!

Anyway, back to my simulation program.  I simulate a bowler using the following basic statistics:  for every 1000 first throws, how many times does the bowler get 10 pins down, 9 pins down, etc.  In addition, each bowler has a spare percentage — that is, of the time that they don’t throw a strike, what are the chances they’ll get a spare?  For simplicity sake, if they don’t throw a spare, I say that all outcomes  are equally likely; for example, if there are 2 pins remaining and their spare percentage is 70%, then they’ll knock down 1 of the remaining 2 pins 15% of the time, and neither of the pins 15% of the time.  I could of course make the simulation more realistic by making those numbers different, but that would make it more complicated and I don’t think it affects too much, especially for top bowlers where they rarely face more than 1 or 2 remaining pins, and rarely miss spares.

Ok, so with a little fiddling I was able to come up with a bowler that appoximates a top PBA bowler.  If a bowler throws strikes 60% of the time, leaves 1 pin standing 25% of the time, leaves 2 pins standing 15% of the time, and has a spare percentage of 80%, then their average will be 223 (based on 100,000 runs of the simulation).  From what I found on the web, the single-season record for average is 228.04, so that’s pretty close.  I also just looked it up and the single-season record for spare conversion percentage is 88.16%.  So 80% might be typical of top PBA bowlers.

As for me?  I estimate my average around 140 (my high is 187).  Someone with the following statistics would bowl an average of 142:  20% strikes, 20% leave 1 pin standing, 20% leave 2 pins standing, 10% leave 7 pins standing, 10% leave 6 pins standing, 10% leave 5 pins standing, 2% for each of 0-4 pins, and a spare percentage of 50%.  Of course there are other combinations that would result in an average of about 140, but that gives a rough idea of what my statistics might be (I don’t *think* I throw gutter balls on the first throw 2% of the time, but it made the numbers work out).

Also, what effect on my average would I have if I got better, improving my strike percentage to 25% and my spare percentage to 60%?  Then my average would go from 142 to 156.  The next time I go bowling (which is admittedly pretty rare), I’ll keep track of these stats and compare them with these numbers.