Thursday, February 11, 2010

Bowling Puzzler III: Syzygy

Anybody who bowls on a team is familiar with the following. Most weeks, somebody has a really good night. Someone else usually has a bad night. From time to time two bowlers on your team will bowl really well. Even more rarely, three bowlers have good nights. For some reason, though, it seems like it is very rare that all four bowlers bowl really well on the same night. It's like the alignment of the planets. It's not that uncommon for a few planets to align in orbit, but to get a true syzygy of all eight (nine?) is a rare thing.

So, here's the puzzle. Assume there is a bowling team that bowls once a week with four members. On any given night, a person can bowl above average or below average. Let's say the chance of doing either is 50%.

How often would you expect all four bowlers to be above average? What about a five person team? How often should all five bowlers exceed their averages?

9 comments:

  1. And NOW I raise mymug to YOU, brother, in homage... You have used syzygy in a bowling blog. Such things are never done by mere mortals... You have the GIFT, young Blogwalker.

    Your question(s) have plagued me since I began bowling (or, at least, began paying attention). My arbitrary scratch gauge of 750 (for 1 game) assumes a 150 average by all 5 bowlers. We have had the potential (really) to have 5 bowlers hit 170 (100 pins better). For some reason, however, this syzygy of which you speak is an extremely rare occurrence-- I'd have to check the logs, but I don't think there's been a game in which we were all over average (I'll look tonight).

    Heck, just getting me over my average in one stinkin' game is struggle enough-- I think the other HCLCers are planning (plotting?) an intervention.

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  2. I knew you'd like that one. I wasn't even thinking about single games. I was thinking about series. Either way, the answer will be the same.

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  3. If my memory of Probability holds, there is a 6% chance with 4 bowlers, 3% chance with 5 bowlers. So, with the HCLC A team, it should happen once each season.

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  4. Doc, once again, you got it. Your memory serves you well.

    For four bowlers, it is solved as: prob= 1/2 * 1/2 * 1/2 * 1/2 = 1/16, or 0.0625. It should happen about once every 16 weeks.

    For five bowlers, just multiply by 1/2 one more time, and you get 1/32 or 0.03125, or once every 32 weeks.

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  5. Once again, thanks for the challenge. E.B. is positioning himself so he can beat the average the rest of this year. We've got a shot.

    On a different note, I'm telling my wife just how healthy bowling is: (http://www.webmd.com/osteoporosis/news/20100208/beer-for-better-bones)

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  6. Very nice but I didn't really need another excuse to drink beer.

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  7. Phew- I'm afraid of statistics quizzes.

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  8. I've been bridge-building:

    http://thumbhole.blogspot.com/2010/02/bowling-puzzler-iii-syzygy-same-topic.html

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  9. Where's the beer-for-better-bowling study?

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