Shortcuts With Chained Probabilities

February 17th, 2021
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.

Comment via: facebook, lesswrong

Recent posts on blogs I like:

Pay For Fiction

Against piracy

via Thing of Things February 29, 2024

When Nurses Lie to You

When the nurse comes to give you the flu shot, they say it won't hurt at all, right? And you trust them. Then they give you the shot, and it hurts! They lied to you. A lot of nurses lie to children about shots and blood draws. Part of it is they probabl…

via Lily Wise's Blog Posts February 28, 2024

How I build and run behavioral interviews

This is an adaptation of an internal doc I wrote for Wave. I used to think that behavioral interviews were basically useless, because it was too easy for candidates to bullshit them and too hard for me to tell what was a good answer. I’d end up grading eve…

via February 25, 2024

more     (via openring)