Shortcuts With Chained Probabilities

February 17th, 2021
math
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 = ~2/1,000,001
This is quite close! Approximating 1 - (1-p)^2 as p*2 was 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^2
If p is very small, then p^2 is negligible, and 2p is 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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