Monday, October 19, 2009

On bowling the same game twice

One of the things that fueled my interest in bowling statistics is a website that can be considered the best mathematical treatment of bowling scores ever written. If you like bowling, statistics, and math, it is definitely worth checking out. One of the first conclusions made on that website is that there are 5,726,905,883,325,784,576 possible games of bowling that can be rolled. That's a big number. If you multiply that number by 12,000 or so, you get the number of stars in the universe.

Given this remarkable diversity, it should be intuitively obvious that the chances of bowling exactly the same game twice are very low. In the simplest sense, if you assume that every game is equally likely, if you were to bowl two games, the probability of them being identical is one divided by that big number. Of course, not all games are equally likely, so the chances should be quite a bit better than that.

Now, I should specify what I mean by "exactly the same game". I'm not talking about two games ending with the same score. This happens quite frequently. What I mean is two games in which every single frame is identical. Determining the actual likelihood of this happening is incredibly difficult to do because certain frame outcomes are more likely than others. For example, if you are a really good bowler, who strikes more than 50% of your frames, then it should be clear that your chances of repeating a game, say a perfect game, should be better than for a bowler who is not very good.

What inspired all of this is a simple thought. I wondered if we had ever done this. Have we bowled the same game twice? We have 312 games in our database. Are any identical? To answer this question requires comparing all possible pairs of games... game 1 vs. game 2, game 1 vs. game 3, game 1 vs. game 4, etc. In all this meant making more than 48,000 comparisons. Each game has 21 possible scoring opportunities, two for frames 1-9, and three for frame 10. For each possible game comparison, I just counted how many out of those 21 scoring opportunities were identical.

The closest we have ever come to repeating the same game was 15, meaning that 15 of 21 scoring opportunities were identical. We have done this twice. To clarify what the hell I'm talking about, look at the two games below. I have highlighted in yellow the portions of the game that are identical. Johnebob rolled one, and Joe the other. Note that frames 1-3 are identical, as are the 2nd half of frame 4 through the end of frame 8. Both bowlers began frame 10 with a strike. That these games ended with nearly identical scores should not be surprising.


Our other example involved myself and Daniele. Here it is with the identical portions in blue:


These games are very similar, but they are still not exactly the same. In fact, both are only 71.4% similar. Given enough time, we will repeat a game, but we may not have enough time in our lives to do so. We have on record 312 games. Every single one is unique. As we add more unique games to our database, the chances of repeating one improves. Think of it like birthdays. As you increase the number of people in a group, the chances of finding a repeated birthday should increase. There are after all only 366 possible birthdays. (In fact, you only need to get 23 people together for the probability of a repeated birthday to exceed 50%. Check out the presidents if you don't believe me.) However, given the immense number of possible bowling games, you have to get a large sample before you have a good shot at repeating a game. I'm not sure how large that sample must be, but I am sure that it is very large.

1 comment:

  1. So, if I haven't already bowled a 300, my chances are pretty good that it may happen the next game? (Teachers really are the WORST students.)

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