Monday, February 1, 2010

The Bell Curve 2 - what looks random maybe isn't

Here is a picture of my hand holding my little basket-like bathroom sink drain filter. I got curious a few years ago, and wondered if I held it like that, gave it a flip like a coin and let it bounce around inside the sink, whether it would seat itself perfectly in the drain hole. I knew it would get close to the drain, because of the shape of the sink -- but would it seat itself perfectly? I thought not: there were too many other ways it could end up.

So I started an experiment. I would flip the thing ten times, and see how many times out of ten flips it would seat itself perfectly. The results surprised me. In 187 trials of ten flips each, the numbers came out to be:

0 "perfect seatings" - 0 occurrences
1 ps - 2 occurrences
2 ps - 6
3 ps - 15
4 ps - 35
5 ps - 51
6 ps - 43
7 ps - 24
8 ps - 11
9 ps - 0
10 ps - 0

Note especially here that the data form a bell-shaped curve which is almost symmetrical. There is a pattern in what looks like randomness or chaos. I won't bore you with the math used to compute the "expected value" of X, the number of perfect seatings out of ten flips, but the answer, given the data, is about 5.18, which means that on average, in general, there is a slightly better than even chance that on any given flip, the strainer will seat itself perfectly.

Here's another example, a game called "coincidences." You shuffle a deck of cards, cut it, and deal thirteen cards off the top; as you lay down the first card, you say "ace"; as you lay down the second card, you say "deuce," as you lay down the third, you say "trey," and so on up to jack, queen, and king. The question is: what is the chance that you'll get at least one "coincidence" as you lay out the cards -- that, for example, as you say "six" you turn up a six? It would seem far-fetched -- until you do it. I played the game 150 times and got:

zero coincidences 54 times
one coincidence 56 times
two coincidences 29 times
three coincidences 9 times
four coincidences 2 times

which means the odds (chance, probability) of getting at least one coincidence are slightly under 2/3; for this group of trials, p(x) = 0.6204. Note "p(x)" means "probability of x happening," where p by definition is a decimal between 0 and 1, where 0 means impossibility and 1 means certainty.

(I mention in passing that this game can be used to win beers at your local watering hole, because the results are counter-intuitive. You never heard me say this.)

And of course a pattern strongly implies a pattern-Maker.

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