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JGWeissman comments on How I Lost 100 Pounds Using TDT - Less Wrong
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No, that would give you the expected number of occurences (assuming independence, which is likely a bad assumption in this case). If the probability of injury in a second is P (given no previous accidents) and you exercise for T seconds, or until an accident occurs, then the probability of an accident occuring during the session is (1 - (1-P)^T), which is the probability of not having the conjunction of T accident free seconds.
I accept the technical correction. But for P close enough to zero, the probability remains nearly linear up to large T. So while you are technically correct I do not think you have refuted my statement as a very close approximation for a typical session lasting an hour or several. Given that, and given the relative difficulty of thinking about an exponential function, I prefer my formula for conveying the essential point in a comprehensible way.
It would be better to make the use of approximation explicit, and to specify the domain in which the approximation is a good one, like "don't forget probability of accident per second is multiplied by time to get approximate total probability, given small total probability".
No no, I am saying I erred outright. I simply used a heuristic and forgot it was a heuristic. You corrected me. That's why I had not identified it as an approximation initially. But that said, I find the correction to make the reader's task difficult, and my heuristic retains approximate validity, so I pointed that out in my reply.