In the infinite limit (or just large-ish x), the probability of at least one success, from nx attempts with 1/x odds on each attempt, will be 1 - ( 1 / e^n )
Cool. Is this right? For something with a 1/n chance of success I can have a 95% chance of success by making 3n attempts, for large values of n. About what does “large” mean here?
95% is a lower bound. It’s more than 95% for all numbers and approaches 95% as n gets bigger. If n=2 (E.G. a coin flip), then you actually have a 98.4% chance of at least one success after 3n (which is 6) attempts.
I mentioned this in the “What I’m not saying” section, but this limit converges rather quickly. I would consider any n≥4 to be “close enough”
e^3 is ~20, so for large n you get 95% of success by doing 3n attempts.
That’s crazy how close that is. e=3√20 (to the nearest half a percent) will be a fun fact that I remember now!
related: https://xkcd.com/217/
Conversely, if you don’t see any success after 3n attempts you have a 95% confidence interval that 0 < p < 1/n (unless you have a strong prior)
https://en.wikipedia.org/wiki/Rule_of_three_(statistics)
I am ordering a motivational poster of this
There are incredibly usefull shorthands. What is the probability for 2n attempts?
In the infinite limit (or just large-ish x), the probability of at least one success, from nx attempts with 1/x odds on each attempt, will be 1 - ( 1 / e^n )
For x attempts, 1 − 1/e = 0.63212
For 2x attempts 1 − 1/e^2 = 0.86466
For 3x attempts 1 − 1/e^3 = 0.95021
And so on
You have the tools necessary to figure this out
Cool. Is this right? For something with a 1/n chance of success I can have a 95% chance of success by making 3n attempts, for large values of n. About what does “large” mean here?
95% is a lower bound. It’s more than 95% for all numbers and approaches 95% as n gets bigger. If n=2 (E.G. a coin flip), then you actually have a 98.4% chance of at least one success after 3n (which is 6) attempts.
I mentioned this in the “What I’m not saying” section, but this limit converges rather quickly. I would consider any n≥4 to be “close enough”