Your Moment in Probability

And in this hypothetical example ALL of your underlying assumptions are essentially correct, the system is both simple and "closed," and the only variance is due to the causal interactions not being known to infinite mathematical accuracy and perfection, giving you the 1% error factor.

The example I like to use is drug or aids or other medical testing. For example, if most of the people tested are clean, then even if the false positive rate is low, then it can be the case that if a given result is positive, it's can actually be more likely to be a false positive than a true positive.

So if you get bad news on a medical test, consider a re-test.

I don't have time to cook up some hypothetical numbers to iliustrate this right now, but I think I may have some handy from my graduate project. I'll search when I have time.

Just run the numbers....

This one's a simple exponential, and your built-in calculator in Windoze can do it easily. I'll do a little rounding to keep the strings short.

0.99¹ = 0.99 = 99%

0.99²⁰ = 0.818 = 81.8%

0.99⁵⁰ = 0.605 = 60.5%

0.99¹⁰⁰ = 0.366 = 36.6%

Even when you're absolutely positive about that 99% accuracy, at the end of 100 iterations your odds of still being correct in your iterated projection are barely better than 1 in 3. But most models aren't even remotely that accurate, so run the lower probability checks. (I'll leave out the first power, that being a bit obvious.) Here's at 95%:

0.95²⁰ = 0.358 = 35.8%

0.95⁵⁰ = 0.077 = 7.7%

0.95¹⁰⁰ = 0.006 = 0.6%

You're down to that barely one in three by twenty iterations. At 90%? Nine-in-ten chance of being on target? Gimme those odds at the horse track! But in an iterative predictive model?

0.90²⁰ = 0.122 = 12.2%

0.90⁵⁰ = 0.005 = 0.5%

0.90¹⁰⁰ = 0.0000027 = 0.00027%, or less than three out of ten million.

This is with a simple iterative predictive model of a closed system running at 99%--95%--90% accuracy. Next time you hear any prediction of anything running out much longer than a dozen cycles, remember this small example. As marghlar so rightly pointed out, this is why the weatherfolks get fuzzier and fuzzier as the time extends.

Your odds are actually slightly better than shown--there is always the slim chance that once the chain of correct answers is broken that a "wrong" answer in subsequent iterations can be right by sheer random accident, thus "resetting the clock," so to speak. But it's not the way to bet.