This rule of 3 may be a good example for reading books and finding typos. It is no where as good for estimating the chances of "something that hasn't happened". It's so random using this rule & claiming the result as an estimate.
Estimating the chances of something that hasn’t happened yet
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Re: Estimating the chances of something that hasn’t happened yet
#92Earlier quoted context omitted.
This is incorrect: there's no reason to expect that X and Y will each appear 3 times in 6 trials if their probabilities are equal. If all 3 measurements of X are smaller than all 3 measurements of Y, then you have X < Y with confidence 1 - 1/8 or 87.5% confidence. You'd need at least 5 measurements to be 95% confident.
You're not considering the right probability space. We have 3 measurements of X and 3 of Y. The question is the distribution on orderings of these six measurements. If X and Y come from the same distribution then all orderings are equally likely.
Ah, I failed to consider that. The original post is correct.
Re: Estimating the chances of something that hasn’t happened yet
#93The rule of three requires quite a lot of assumptions about the nature of the phenomena. Or as Taleb says it: > Consider a turkey that is fed every day. Every single feeding will firm up the bird's belief that it is the general rule of life to be fed every day by friendly members of the human race "looking out for its best interests," as a politician would say. > On the afternoon of the Wednesday before Thanksgiving,…
It certainly does not require unwarranted assumptions, the proposed approach is consistent with the well-known turkey scenario.
From the experience of feeding a turkey can infer that being slaughtered is a rare event, and that the likelihood it happening exactly tomorrow (without having access to a calendar) is not necessarily 0, but is below a certain rate - and it's definitely not likely to happen three times in the next week. Surviving for 180 days is reasonable justification to assume that, on average, turkeys get slaughtered less frequently than every 60 days, i.e. that the likelihood of Thanksgiving suddenly arriving tomorrow isn't 50% but rather something below 2%.
Re: Estimating the chances of something that hasn’t happened yet
#94Earlier quoted context omitted.
The rule of three gives an upper bound estimate that each page has a 3/200 or 1.5% chance of having a typo. For the 20,000 page book, the probability of typo per page is still 1.5%, but you have to roll the dice 19,800 more times, each time you turn a new page. The formula for the upper bound on probability of getting through all the remaining pages without a typo is 1 - (1-3/200)^19800 Which of course is basically 1…
Minor correction to your otherwise excellent final statement: "... or equivalently the total number of [ pages with ] typos in the 20,000 page book is between 0 and 300." We're not measuring the number of typos on a page, but only whether the page contains a typo or not. So we can't speak about the number of typos.
The math in that article is based on the assumption that typos are scarce, because that's the situation where it makes sense that you've seen 0 events. If that probability is close to zero, the probability of two typos per page is insignificantly small and can be discarded, and the expected number of typos in the book is approximately equal to the expected number of pages with typos. So in the context of these assumptions it's not necessary to consider this difference.
Just as the article itself states regarding the approximation "Since log(1-p) is approximately –p for small values of p" which relies on discarding p-squared terms as insignificantly small.
Re: Estimating the chances of something that hasn’t happened yet
#95Earlier quoted context omitted.
Yeah this is weird. Also, if OP doesn't speak Italian themselves, how can they attest to the quality of the translation?
I have a couple L2’s that I mostly speak and read, virtually no writing - my grammar is poor and my vocabulary is limited. Because of this I would feel uncomfortable/embarrassed to translate any tech stuff I wrote to it. However, I could read someone else’s translation and know if it’s way off the mark. Also, I have plenty of friends that speak the language natively, so I could ask them to review. If they say it’s go…
Oh ok cool :)
Re: Estimating the chances of something that hasn’t happened yet
#96So, according to this rule, if I wait for 10 minutes for a bus to come and none does, and then I wait for another 10 minutes for an alien invasion and none happens, the two have the same upper bound on their probability? Or are we going to start talking about priors, on buses and alien invasions, in which case the rule of three is not really useful? If I want to know how likely a specific book is to have typos, can't…
This is the scenario where you know nothing else other than you waited 10 minutes for this event and it didn't happen.
Sure, if you have more data, then you can get much tighter bounds for your estimate.
Re: Estimating the chances of something that hasn’t happened yet
#97Oh, numeric astrology and probability tantras.
Future is not predictable by definition. It is just an abstract concept, a projection of the mind. Any modeling, however close to reality it might seem to be, is disconnected from it, like a movie or a cartoon. Following complex probabilistic inferences based on sophisticated models is like to act in life guided by movies or tantric literature (unless you are Goldman Sachs, of course).
For a fully observable, discrete, fully deterministic models, such as dice or a deck of cards probability could only say how likely a certain outcome might be, but not (and never) what exactly the next outcome would be.
Estimation of anything about non-fully-observable, partially-deterministic environments is a fucking numeric astrology with cosplay of being a math genius.
No matter what kind of math you pile up - equations from thermodynamics, gaussian distributions or what not it is still disconnected from reality stories, like the ones in tantras.
Re: Estimating the chances of something that hasn’t happened yet
#98Earlier quoted context omitted.
> It's understandable to not mention this I feel like the very first word, "Estimating" made what you're talking about pretty clear. It's extremely common for estimates to make simplifying assumptions that don't necessarily hold if you care about accuracy or specific situations. He also called the rule of three "quick and dirty". > If you were to randomly read 20 pages in a book and find no typos, 15% probability mak…
Interesting that you would worry about repeats. Most people would interpret "randomly read 20 pages" as "choose 20 from n" not "take 20 rand(n)".
You're right, and mathematicians and computer scientists aren't most people. We are being pedantic / precise here. If someone wants to caution that people need to be really careful about estimates because there are some assumptions involved, then it's fair to hold them to their own standard, isn't it? If typos are uniform, then the suggestion to read random pages doesn't help. And whether or not people interpret it one way most of the time, if you interpret "random" like a programmer, you've introduced a small amount of bias, right?
Re: Estimating the chances of something that hasn’t happened yet
#99So, according to this rule, if I wait for 10 minutes for a bus to come and none does, and then I wait for another 10 minutes for an alien invasion and none happens, the two have the same upper bound on their probability? Or are we going to start talking about priors, on buses and alien invasions, in which case the rule of three is not really useful? If I want to know how likely a specific book is to have typos, can't…
This is the scenario where you know nothing else other than you waited 10 minutes for this event and it didn't happen. Sure, if you have more data, then you can get much tighter bounds for your estimate.
This is the scenario where you know nothing else other than you waited 10 minutes
To make the math work, I think you need to make several other assumptions. Don't you also have to know that that ten minutes you sampled are representative? That the events (if they did occur) are independent? It seems odd that you'd in a situation where you can rely on specific assumptions like these while also believing that buses and aliens are equally likely to appear.
Re: Estimating the chances of something that hasn’t happened yet
#100Mildly interesting trivia: there's a Chinese idiom stating "compare 3 shops for a good deal" (貨比三家不吃虧), I guess that makes sense from a statistical standpoint. :)