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?
Referenced in: Peekskill Lyme Incidence
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.
I went through this exact chain of reasoning when I started using the microCOVID system.
yeasssssss
if we hadn't had this "wait a minute, you can add these" realization, treating probabilities as points and budgeting them would've *never happened*, like quite plausibly the whole project wouldn't have been a thing
If you do it twice, your risk of death is one in a million. The reason for this, is that if you do it twice, it's clear that you did not die the first time, so we only have to be concerned about whether there was any risk of death the second time.
I think Jeff's talking about planning, rather than after the fact.
Eg. I want to bring these mushrooms to Grandma's, but there's a two in a million chance that a wolf will eat me on the way there (when I'm carrying the food) and a one in a million chance that he'll eat me on the way home, so my overall chance of being eaten is three in a million
Marcus My comment was half just a joke, and half about why the person asking the question should be more careful in how they pose it.
This is sort of like the old rule that all mushrooms are edible, but some mushrooms are only edible once.
Fair enough.
Probability is more complicated anyway, so you should be reevaluating your odds based on data. The classic example of "If a fair coin lands heads 75 times in a row, what's the chance will get heads on the 76th toss?" is a bad, imo. It's not likely to be 1/2, since you need to reevaluate your assumption that it was fair.