Wednesday, April 21, 2004


I don't understand how probability calculations are supposed to apply to real life, if at all. I also don't understand how probability applies after the fact, if at all.

1. The Real Life Issue: A CT scan might be 90% accurate at detecting a hemorrhage. Standard probability theory would just say, "Well, do two tests, and voila, you have 99% accuracy, rather than 90%." I.e., you multiply the 10% error rate of the first test by the 10% error rate of the second test. The end result: A 1% error rate.

But that assumes that all results are random. This cannot possibly be true. If a hemorrhage had happened that was of the wrong size or in the wrong place, the CT scan might not have a 10% error rate in the first place. In fact, it might be guaranteed that the CT scan would fail to pick up the hemorrhage, in which case the error rate would be 100%, no matter how many scans were performed.

This is just an example, and I might have the facts wrong. But the theory is correct. Another hypothetical: You have a 5% chance (I'm making up the number) of getting into an auto accident this year -- but that is an aggregate calculation that cannot possibly take into account the fact that you drive with exceeding caution, never speed or drink, etc. Your real chance is nearer to zero.

Or whatever. My point applies everywhere. Probability calculations depend on randomness. Yet -- even apart from the difficult issue of determinism -- nothing in the real world is really random.

2. The After-the-Fact Issue: How on earth does probability apply after the fact? You have a 5% chance of getting into an auto accident this year -- but in fact, you just had an accident this week. Does the 5% figure still even mean anything? Did you always have a 5% chance, and you were just unlucky enough to fall into the 5%? Or did you personally have a 100% chance (a chance that applies to only 5% of people ahead of time)?

Alternatively, you never have an accident this year. Did you have a 5% chance that left you in the 95% category? Or a 0% chance given that it never happened?

I'm sure someone has written on these questions in depth, but I don't know who.


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