Let's say you're considering an activity with a risk of death of one in a million. If you do it twice, is your risk two in a million?
Technically, it's just under:
1 - (1 - 1/1,000,000)^2 = ~2/1,000,001This is quite close! Approximating
1 - (1-p)^2as
p*2was only off by 0.00005%.
On the other hand, say you roll a die twice looking for a 1:
1 - (1 - 1/6)^2 = ~31%The approximation would have given:
1/6 * 2 = ~33%Which is off by 8%. And if we flip a coin looking for a tails:
1/2 * 2 = 100%Which is clearly wrong since you could get heads twice in a row.
It seems like this shortcut is better for small probabilities; why?
If something has probability
p, then the chance of it
happening at least once in two independent tries is:
1 - (1-p)^2 = 1 - (1 - 2p + p^2) = 1 - 1 + 2p - p^2 = 2p - p^2If
pis very small, then
p^2is negligible, and
2pis only a very slight overestimate. As it gets larger, however, skipping it becomes more of a problem.
This is the calculation that people do when adding micromorts: you can't die from the same thing multiple times, but your chance of death stays low enough that the inaccuracy of naively combining these probabilities is much smaller than the margin of error on our estimates.
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