The Hand rule is a funny little interloper from the world of thermodynamics. The burden on a reasonable person to take precautions should not be greater than the loss that might occur without the precaution times the probability of the sort of accident that would cause the loss. This looks eerily familiar to a student of entropy, but the logarithm is missing. Except that the log should be included because otherwise the probability of any given accident becomes both vanishingly small, and incomparable to the probability of any other loss. Like calculating the actual entropy of a system, it is impossible to calculate the actual probabilities of any given accident, so the formal mathematical expression is moot. But the rule really should be the loss times logarithm of the probability so that different losses can be compared proportionally, and because of the way probabilities diminish exponentially as possibilities increase. Take, for example, the probability of an accident occurring with a toothbrush, versus a bandana. Because the toothbrush has only one real use, you can only foresee a few types of accidents with it, but a bandana could be used anywhere, so that all kinds of remotely possible accidents become foreseeable. For the burden of care for these two products to be comparable, you have to use the logarithm of the probability of the foreseeable accidents.

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