The Triangle

Issue date: 4/8/05 Section: Sci-Tech

Media Credit: Karen Maziarz

I insert a dollar bill in the vending machine to buy one chocolate bar, but surprisingly (and happily) find that the generous machine gives me two instead! And it cannot take the mistake back anymore. Since I haven't in a 1000 years imagined I could obtain two bars at half the price, I was more than happy to accept this extra gift, only with worries that next time, I'll be given none.

This rarely happens, but that's not equal to never! Just because we wait our whole long lives for the happy surprise once only ­- too hopeless to be true ­- we grant this impression to be (incorrectly) 0.

There may be errors or imprecisions here and there when we try to understand the world, but we are just too negligent to notice that. Look at the following example:

You are flipping a coin (to decide whether to turn down Mike, for example), and of course you know that the probability of Mr. Washington appearing on top is equal to that of Mr. Eagle appearing upside down; it's exactly one half. And you're even smart enough to know that after you've gained one Mr. Washington, the probability that you get a second Mr. Washington is still exactly one half.

However, the image that you have flipped ten thousand times may result in 5000 Mr. Washingtons and 5000 equal Mr. Eagles. Then after one Mr. Washington has appeared, the probability to have another Mr. Washington should be only 4999/10000, which is slightly different from the supposed one half! How come?

Well, 1/10000 may be too small a gap for 4999/10000; so small that we can ignore it ­- that explains it!

But science is exact.

In other words, the "second time Mr. Washington probability" question is a single-choice one, and the smart you can even guess the answer can be no other. So, what about the gap?

OK, let's do the flipping again, but 100 times instead. If you think Mr. Washington and Mr. Eagle comes out 50 times exactly each, then the gap would be enlargened to 1/100. Now, you see, the gap depends on number of samples directly. And if we just flip 10 times, the gap can be as large as 1/10, which means, this way is not valid to show the equal distribution.

Also invalid are 100, 1000, 10000 times ...

So, the mistake is brought in when we regard that the 10000 time flipping can exactly represent the distribution. But something you should notice is when flipping numbers increase, the gap is decreasing. When the gap is 100% bridged, the sample numbers have to go as crazy much as infinity!



Yuanxin Sun is a first year graduate student in physics.

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