## Shortcuts With Chained Probabilities |
February 17th, 2021 |

math |

Technically, it's just under:

1 - (1 - 1/1,000,000)^2 = ~2/1,000,001This 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^2If

`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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