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Subject: A bowler/statistician's take on Bowl and Score rss

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South Euclid
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When I was a kid I was an avid bowler even before I could throw the ball properly, so it’s no surprise that my parents bought me Bowl and Score. I quickly realized that the game had no skill component whatsoever. Still, once in a while I liked to toss the dice a few times, to see how many strikes I could string together. I forgot all about Bowl and Score for ages, but I recently encountered it on a BGG thread, and I got to thinking: just how good a bowler is Bowl and Score? Better than I?

Briefly, the game comprises ten six-sided dice with a bowling pin printed on one face of each. The dice are treated like pins: roll the ten dice, and those with the pin side up are “left” after the first ball. These are thrown a second time for the spare shot. The dice are thrown in the same way for every frame of a standard game of bowling, including extra rolls as needed in the tenth frame. Scoring—at least in the version I played—is exactly as in real bowling.

You can find Bowl and Score’s average to a reasonable degree of accuracy by rolling a few hundred games and taking the mean of the results. But to a statistician, just finding the average isn’t enough. To really understand the game, you need to characterize it completely, by finding the full probability distribution: the likelihood of every possible score, between 0 and 300.

Working out all the probabilities mathematically is straightforward in theory but very, very tedious in practice. (How many ways are there to score, say, 157? Care to enumerate all of them?) It’s much easier to approximate the probability distribution by sampling. Thanks to modern computers, random numbers are available in bulk. If we simulate enough games we can approximate the true distribution as closely as we wish, constrained only by the number of CPU hours we can stand to invest. A couple nights ago, after drinking a little too much Dr. Pepper, I decided to program a Bowl and Score simulator.

Bowl and Score was easy enough to implement. Using the statistical programming language R, I wrote the algorithm in only ten minutes, and simulated 100,000 games of bowling in another ten, on my not-too-new desktop PC. Below is a histogram of the results.

Overall, the distribution of scores looks like a normal (bell-shaped) curve with a sine wave superimposed on it. The waviness is realistic: the mechanics of bowling dictate that some totals are easier to get than others in the same neighborhood. There is also a long upper tail of high games, when Bowl and Score was apparently “in the groove.” We think of amazingly good and bad performances in sports as mostly a psychological thing, but in fact, chance accounts for quite a lot, as well.

As shown on the graph, the average (mean) score was 164.5. It is customary in bowling to round averages (and handicaps) down, so officially, Bowl and Score is a 164 bowler.

Bowl and Score is pretty consistent: only 1% of its games are 126 or below, and only 1% are 204 or higher. (The standard deviation, the usual measure of variability, was 16.6.) Only 2.0% of the games broke 200, a common criterion for a “good game”—one for which most automatic scorers will congratulate you. The lowest and highest scores of the 100,000 games were 98 and 265, respectively. The 265 game had spares on the first and fourth frames, and strikes everywhere else except for the very last throw, which scored 8. Choke!

This score distribution is actually realistic for a certain type of bowler: a good straight-ball bowler. As beginners quickly discover, it is difficult to throw strikes with a straight ball: the window of opportunity, in which the ball hits “high” enough on the head pin to carry the 5-pin, but not so high as to leave the 4 or 6, is much narrower than for a curve ball. Indeed, Bowl and Score rolls on average only 1.6 strikes per game: the probability of rolling a strike is 0.162, or 1 in 6.2. Good straight-ball rollers harvest most of their pins by shooting spares accurately.

My average topped out around 150 with a straight ball, and didn't improve until I learned to throw a hook. The best straight-ball shooter I’ve ever seen had a 175 average. She didn’t bowl any more strikes than Bowl and Score does, but she never missed a spare.

(Edited to fix the numbers in the following paragraph.)

Swapping the six-sided dice for dice with more sides would, of course, improve the scores. Using a d8 instead of a d6 only raises Bowl and Score’s average to 181 (high score of 10,000 games = 259), and using a d10 to 194 (high score of 10,000 games = 279). Neither of these bowlers is even close to going pro: the pros typically average 220 under trying circumstances. Even using a d12, Bowl and Score can only scrape together a 204 average.

In any case, I was happy to discover that I am, in fact, currently a better bowler than Bowl and Score.
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Joseph Betz
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Nice breakdown of stats here.I have only played this a few times and have a high game of 194 or something like that.I think I had 5 strikes in that game.It would probably be easier to bowl a perfect game in real life than in this game.A score of around 150 or so does seem to give you a good shot of winning this game and your stats seem to back that up.
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