User:IssaRice/Additivity of small risks

Suppose you want to engage in some risky activities such as driving on the highway or swimming in the river, which have some small probability of resulting in death. It turns out that when the probabilities involved are independent and small, one can simply add the probabilities together instead of doing the more complicated correct calculation.

Let $p_{1},\dots ,p_{n}$ be the probabilities of the risk of dying from each of activities $1,\dots ,n$, where each $0\leq p_{i}\ll 1$. Then the probability of dying from doing all activities is $\Pr(\mathrm {death} )=1-\prod _{i}(1-p_{i})$. This happens because for independent probabilities we know how to take conjunctions (AND) by multiplying but not disjunctions (OR) so we must first invert and use de Morgan's laws then invert again.

But now let $m_{i}:=-\log(1-p_{i})$. Then we have $\Pr(\mathrm {death} )=1-\exp(-\sum _{i}m_{i})$. When each $p_{i}$ is small, we have $m_{i}\approx p_{i}$. Thus $\sum _{i}m_{i}$ is also small. For small positive values of $x$ we have $1-\exp(-x)\approx x$. Thus we have $\Pr(\mathrm {death} )=1-\exp(-\sum _{i}m_{i})\approx \sum _{i}m_{i}\approx \sum _{i}p_{i}$.

If $y=-\log(1-x)$ then $x=1-\exp(-y)$ which means that the two functions are inverses of each other. So it makes sense that when either quantity is small, the other is also small.