Thursday, April 1, 2010

Bowling Puzzler X: The Dutch 200 Game

Does anybody know why paying individually on a date is called "going Dutch"? Likewise, what is the etymology of the Dutch 200 game? These are interesting puzzles in and of themselves. I'm going to send EB down that path if he is interested in continuing his work on Holschuh's Etymological Dictionary of Bowling.

Anyway, today's challenge is about the Dutch 200 game. According to last year's USBC Rulebook, a Dutch 200 Game is "A game of alternating strikes and spares with a game total of 200." If you were fortunate enough to bowl one of these last year, you would have received a patch for it. However, the special achievement awards appear to have been eliminated from the most recent rulebook.

A game of alternating strikes and spares will always result in a score of 200 because 20 pins will be garnered for each frame, and it does not matter whether the first frame is a strike or a spare. This is a very rare occurrence in bowling, and today's puzzler concerns that rarity. Let's assume that we are dealing with a very skilled bowler. So, assume this:

1. The probability of getting a strike on a given frame is 50%.
2. If that bowler does not get a strike, the probability of picking up a spare is 75%.

For a single game bowled, what is the probability of this bowler getting a Dutch 200 game? Or put another way, how many games on average would have to be bowled for one of them to be a Dutch 200 game?

[Good luck. You're gonna need it!]

Click on the icon to the right for the answer.

13 comments:

  1. I should have paid more attention in maths class.

    My guess is: The probability of this bowler getting a dutch game is about 0.37% or one in 270 games.

    How did I get to this? I multiplied the probability of a strike with the probability of getting a spare (0.5 * 0.75 * 0.5 etc)

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  2. Well, you are on the right track, but you are not correct.

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  3. Here is a hint. For any frame, there are three possible outcomes: strike, spare, and open. In this system, the probability of a strike is given. It is 50%. What is the probability of a spare? In your answer, you assume it is 75%, but it is not. If you have a 50% chance of getting a strike, then the probability of getting a spare has to be less than 50%, right?

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  4. And I should clarify that. The probability of picking up a spare is 75%. That's not the same thing as the probability of getting a spare on a given frame.

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  5. OK, 2nd attempt:

    I'm using 0.5 * 0.375 * 0.5 etc where the 0.375 comes from 75% of 50%.

    The probability of this bowler getting a dutch game is about 0.01158% or one in 8635 games.

    Don't know if this is right either, he has a better chance of getting a 300 game (0.02441%).

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  6. Alright, now we're getting somewhere. You are a lot closer but still not correct.

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  7. Is this your idea of a cruel April Fool's Day joke?

    I'm not EVEN going to hazard a guess this time, however I defer the etymology issue to Wikipedia, which conjectures that "Dutch 200" has something to do with "Dutch treat" (each person pays their own way), but nothing to do with "Dutch wife" or "Dutch courage," both of which might be found in a bowling alley near you.

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  8. I wish. It has just become a Thursday tradition. I figure I will do it til the end of the season. That's four more weeks of puzzles.

    But I know how much you love words, and I knew you'd come through for me. Now, you have two entries in your bowling etymology dictionary.

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  9. Ok. Here's another hint. There are two ways to get a dutch 200 game.

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  10. Ok you bums, I added the answer to the post. At least MD gave it the old college try.

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  11. I was going to do the calculation for the 2nd possibility and then average the 2 results; I would have still been wrong.

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  12. At least you gave it a shot. In probabilities, if the question asks the probability both of two events occurring you multiply. If it asks for the chance of one or another thing occurring, you add.

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