Comments on: The stats of bowling 300

Question: The stats of bowling 300

roofus — Mon, 19 Feb 2007 15:14:14 -0800

ProbabilityFilter: Can I predict, from my bowling average, how many games it will be (on average) until I bowl a perfect 300 game?

There must be a fixed, but non-linear, relationship between bowling average and likelihood of bowling a perfect game. Assuming I don't choke on the 12th strike, can I work out how many games it will take me to achieve perfection? My maths skills have failed at the stage of trying to convert my average into a strike percentage.

By: noloveforned

noloveforned — Mon, 19 Feb 2007 15:29:03 -0800

well, if you can pretend each roll of the bowling ball is independent, just take your percentage of strikes and raise it to the 12th power. that's the probability that twelve successive rolls will be strikes. 1 divided by this number would be the average number of sets of 12 rolls it'd take...

By: magikker

magikker — Mon, 19 Feb 2007 15:30:32 -0800

I'd say take a bell curve and check how many standard deviations a perfect game is from your mean. basic stats from there out...

Though Noloveforned's answer is better

By: Doofus Magoo

Doofus Magoo — Mon, 19 Feb 2007 15:30:41 -0800

Mathematically speaking, I doubt there's a way to do it extrapolate your odds of rolling a perfect game from your average; you'd have to know your "strike rate" (i.e., the percentage of times you hit a strike). Your "odds" of bowling a perfect game (ceteris paribus) are that strike rate to the 12th power.

The reason I say it probably can't be done from your average is that a 180 average could mean several different things -- you never hit a strike, but can pick up a mean spare, or that you're more erratic, hitting a strike or open frame every frame. The second type would obviously (?) have a better chance at some day rolling a perfect game.

By: jellicle

jellicle — Mon, 19 Feb 2007 15:31:19 -0800

If you have an 80% chance of throwing a strike in any given frame, your odds of a perfect game are (0.8)^12 = 6.8%.

70% strikes: (0.7)^12 = 1%
60% strikes: (0.6)^12 = 0.2%
50% strikes: (0.5)^12 = 0.02%
40% strikes: (0.4)^12 = 0.0016%
30% strikes: (0.3)^12 = 0.000053%
20% strikes: (0.2)^12 = 0.00000004%
10% strikes: (0.1)^12 = 0.000000000001% <--me

So if you're throwing at least 7 strikes per game, hang in there, it'll happen eventually. If you aren't, don't.

By: noloveforned

noloveforned — Mon, 19 Feb 2007 15:32:03 -0800

ps - some quick math, if you roll 90% strikes, there's a 25% of rolling 12 in a row, which works out to roughly a perfect game every 4 games. 80% strikes drops it down to under 7% (15 games) and 70% is roughly 1% (100 games).

for those of us that manage 20% strikes, it'll only take about 250 million games to hit 300.

By: CrunchyFrog

CrunchyFrog — Mon, 19 Feb 2007 15:32:51 -0800

Well, if we assume that your odds of getting a strike are some fixed number, p, then your odds of bowling 12 consecutive strikes are p^12.

If you don't have the odds of one single throw being a strike, but you have calculated the your average and standard deviation, you can make an educated guess in a different way: use the normal distribution. (The Central Limit Theorem suggests that this is a pretty good approximation.)

The method for using the normal distribution can be found an introductory statistics book.

By: Blazecock Pileon

Blazecock Pileon — Mon, 19 Feb 2007 15:34:51 -0800

Logarithms of five-pin bowling scores are normally distributed.

You could test if your ten-pin bowling scores will fit along a log-bell curve (a logarithmic "normal" or "Gaussian" probability distribution) with Excel.

By normalizing your log-score histogram and finding a mean and variance, you can use Excel to measure the likelihood of hitting a 300 or greater.

By: deadmessenger

deadmessenger — Mon, 19 Feb 2007 15:36:09 -0800

Although it seems counterintuitive, for averages below 200 or so, you can't really make a prediction that way.

I used to bowl competitively (190 avg over a couple of seasons). Speaking generally, if you were to compare 2 bowlers, one with a lot of power but poor accuracy on spares (and thus a lot of open frames), and a "finesse" bowler who didn't roll hard but almost never missed a spare, the "finesse" bowler would (generally) have an average 30 points higher than the muscle bowler, despite making many fewer strikes per game.

For example, one can bowl a 189 without rolling a single strike. (9 on the first ball, spare, repeat for all 10 frames). This happens pretty often - I've done it at least 3 times that I can remember. On the other hand, someone who strikes every other frame starting in the first, but rolls a 9-miss in the 5 other frames will end up with a 140 to show for their five strikes and five open frames. Unfortunately, I've bowled a game like this, as well.

By: jtfowl0

jtfowl0 — Mon, 19 Feb 2007 15:37:00 -0800

If you recorded scores of enough games, you can calculate your mean score its variance/standard deviation. Figure out how many standard deviations 300 is above your mean, and you can see how rare a 300 game would be fr someone with an average equal to yours. You probably need to have records for 30-50 games for this to be a decent estiamte.

For instance, if you averaged 225 a game with a standard deviation of 25, a 300 game would be 3 standard deviations above the mean, which you would expect to see less than .15% of the time, or 1 time every 667.

There are certainly some problems with this analysis. First, it assumes that your bowling scores are distributed normally, which I'm not sure sure about. Second, it implies that your bowling scores are randomly distributed around your mean score and are unrelated to each other, the latter of which I'm 100% sure is not true. Nonethess, as a really rough approximation, its not completely horrible and is much less complex than doing it more accurately.

By: jtfowl0

jtfowl0 — Mon, 19 Feb 2007 15:38:30 -0800

Or on preview, what all the other guys said.

By: deadmessenger

deadmessenger — Mon, 19 Feb 2007 15:39:38 -0800

Correction - one can bowl a 199 without rolling a single strike.

By: roofus

roofus — Mon, 19 Feb 2007 15:43:01 -0800

By: SpookyFish

SpookyFish — Mon, 19 Feb 2007 15:43:31 -0800

A ton of info on bowling probabilities, with graphs, excel sheets and more.

By: deadmessenger

deadmessenger — Mon, 19 Feb 2007 15:44:59 -0800

Also, anecdotally, I bowled in a scratch (no handicap) league, where most bowlers had an average over 190, and a league average somewhere around 200. In the 2 years I was in that league, I witnessed 4 perfect games. There were 24 teams in the league, 6 people per team, and 32 weeks in the season. Some quick arithmetic says that that means 4,608 games per season, for a total of 9,216 games, or a perfect game every 1152 games for bowlers averaging around 200.

Purely anecdotal, but it sounds about right to me.

By: muddgirl

muddgirl — Mon, 19 Feb 2007 15:45:03 -0800

deadmessenger - check your math again. I'm pretty sure the highest one can bowl without a single strike is 190 - 19 points per frame x 10 frames.

By: Doofus Magoo

Doofus Magoo — Mon, 19 Feb 2007 15:48:26 -0800

muddgirl: I'm pretty sure the highest one can bowl without a single strike is 190 - 19 points per frame x 10 frames.

Nine-spare-nine in the tenth frame? It's been a while since I bowled, but you get a third ball in the tenth if you pick up a spare, right?

By: muddgirl

muddgirl — Mon, 19 Feb 2007 15:49:32 -0800

9-spare-9 in the 10th frame is still only 19 points. That third ball only counts once, unfortunately :)

By: monkeymadness

monkeymadness — Mon, 19 Feb 2007 15:49:48 -0800

If you don't have all the data on how many strikes vs. spares you've picked up, (and who does, I only remember when I get to do the turkey dance), you can get a quick and dirty number by just taking:
(your average)/300 and calling it your "Success Percentage" or something like that, and just consider that your probability of striking on any roll. Then follow what noloveforned said: 1/SP^12. Not especially useful, but best thing you can get without all the work.

By: deadmessenger

deadmessenger — Mon, 19 Feb 2007 15:49:50 -0800

Muddgirl - we're both wrong, actually. It's 19 each for the first 9 complete frames, but 28 for the tenth. (19 for the complete 10th frame, and 9 for the bonus third ball).

By: deadmessenger

deadmessenger — Mon, 19 Feb 2007 15:58:20 -0800

@muddgirl: Actually, you're right. Now that I've sat down with a scoresheet instead of trying to work it out in my head, 190 is the magic number for a game full of 9-spares.

That'll teach me to try to post at the end of a long workday.

/slinks back off

By: roofus

roofus — Mon, 19 Feb 2007 15:59:49 -0800

By tinkering with this spreadsheet from Spookyfish's link, I came up with the answer that I have somewhere between a 1 in (1/(5.31441 × 10-7)) chance and a 1 in (1/(1.6777216 × 10-5) chance of bowling a perfect game. I might just quit bowling now.

By: Blazecock Pileon

Blazecock Pileon — Mon, 19 Feb 2007 16:05:46 -0800

By tinkering with this spreadsheet from Spookyfish's link, I came up with the answer that I have somewhere between a 1 in (1/(5.31441 × 10-7)) chance and a 1 in (1/(1.6777216 × 10-5) chance of bowling a perfect game. I might just quit bowling now.

The great thing about statistics is that you might get lucky and beat the odds.

By: geoff.

geoff. — Mon, 19 Feb 2007 16:11:12 -0800

I don't think bowling scores are anything like normally distributed, because the scoring system rewards chains of strikes. For an average bowler the distribution has positive skewness, and probably low kurtosis too.

I am going to have to strongly agree. Each roll is not going to be independent as we must account for physical fatigue and whatever marginal skill is gained from each roll.

You can also cheat the system, the frame in which you do not get a strike you can hypothetically quit and start over until you bowl the necessary chain of strikes to achieve a 300 score.

I do not believe it is possible to aggregate data and come up with a solution through probability theory.

By: fogster

fogster — Mon, 19 Feb 2007 18:22:06 -0800

Short answer, no. While I don't count myself a professional, or even worthy of giving advice, I've bowled since I was a kid, and have worked in a bowling center for the past few years when I'm not away at school.

There's obviously a positive correlation between one's bowling average and how many 300 games they've bowled, but, beyond that, I don't think you can extrapolate anything. A few comments:

- Much of the difficult is mental pressure, as most people who've come close (or gotten it) will attest. I don't have anything to back it up, but I feel like, as your number of strikes in a row increases, the probability of you striking on your next ball decreases greatly.

- The fit between average and 300s isn't 'tight'--there are people who've rolled a 300 who really aren't great bowlers, and I also know people who've been bowling (very well) for decades who are yet to hit 300.

Any analysis of the 'odds' has got to take into account someone's average: my odds of rolling a 300 are higher than someone who doesn't bowl regularly, but way lower than some of the guys I've bowled with. But even then, I don't think average can predict the odds of a 300 too reliably.

By: Hildago

Hildago — Mon, 19 Feb 2007 19:38:48 -0800

The other thing is that if you're an on-average 180 bowler and you bowl enough games to "statistically" hit a perfect game, your average probably won't be 180 any more.

By: ikkyu2

ikkyu2 — Mon, 19 Feb 2007 19:58:04 -0800

One way to analyze this problem: your mean score is a measure of central tendency of a distribution (in this case, the distribution of bowling scores over many games). However, bowling a perfect game does not occur at the center of the distribution of games; rather, it occurs far out on the tail of the distribution.

Statisticians know that measures of central tendency help you describe the center of a distribution - in some cases. However, they're notoriously unhelpful in describing the tails of those distributions.