I don't know the first thing about schools of thought in statistics (frequentist? Bayesian?) but something feels fishy about extrapolating a probability based on sample size (or number of trials) alone. 20 pages, no typos, Would someone care to explain why the math says it shouldn't matter (for reasonably small values of p)?
Estimating the chances of something that hasn’t happened yet
131–140 of 155 posts
Re: Estimating the chances of something that hasn’t happened yet
#132I wouldn't use a probability density of typo's per page, but instead the probability that a word is spelled wrong.
Then it's like drawing marbles from a vase containing an unknown proportion of blue and red marbles.
I would use the formula (M+1)/(N+2) where N is the number of words, and M is the number of mistakes. Note that for a large corpus (M+1)/(N+2) approaches -> M/N, so we recover the frequentist probability.
Also note the author (John D Cook) correctly expresses intuitive doubt that 0 typos / 20 pages can not be construed as certainty of no mistakes. Similarily, seeing a mistake in every word in a subset does not guarantee that all words in the book will have a typo. Let's look at the modified formula (M+1)/(N+2) in these cases: if we observe no typos (M=0) in 1000 words, it estimates the typo probability as (0+1)/(1000+2)=1/1002 != 0, so we can't rule out mistakes. Similarily if all words were typos (M=N) for the first 1000 words we get 1001/1002 != 1, so we can't be sure every future word is a typo.
Check out Norman D Megill's paper on "Estimating Bernoulli trial probability from a small sample" where the formula (M+1)/(N+2) is derived, on page 3 it appears:
Re: Estimating the chances of something that hasn’t happened yet
#133Earlier quoted context omitted.
My personal favourite quick-and-dirty trick: A quick way to estimate any distribution's median is to draw 5 random samples. There's a >90% chance that the median is between the biggest and the smallest value.
The final bit is complete bunkum for an unknown distribution. Whether the distribution is monotonic and if not, how symmetric, is the major determinant of occurrence of such result. Giving a flat number is completely bogus.
I don't understand what you mean. It doesn't actually matter what shape the distribution takes. That figure is derived entirely from what the "median" means, i.e. a random sample has 50% chance of being greater than the median, and 50% chance of being lower than it.
A bit of maths would show that the probability that all 5 samples lie entirely above or entirely below the median (i.e. the median is NOT between the greatest and the smallest value) is 1/16. That gives you a 15/16 (= 93.75%) chance that the median is contained within the bounds.
Re: Estimating the chances of something that hasn’t happened yet
#134I've seen a very similar problem to this referred to as "black swan events"[1]. The whole point is you can't actually compute it. You see the period, it's there. What he's doing here isn't science, it's a guess. As others have pointed it, it's rather rare that events happen with perfectly distributed probability (probably the opposite). For instance, the chance of being in an accident is much higher if an accident ju…
>The whole point is you can't actually compute it.
You can't compute the probability. But you can compute an upper bound on the probability with a reasonable confidence, which is what the author is doing here.
This is standard statistics and might be useful in some cases. It does assume that the events are independent, but this is a pretty standard assumption that you have to check whether it applies in your case, or at least approximately applies in your case.
>What he's doing here isn't science, it's a guess.
Which is a pretty big thing in this subfield mathematics called statistics.
Re: Estimating the chances of something that hasn’t happened yet
#135Earlier quoted context omitted.
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.
You always have more data -background knowledge- unless you've only existed in those last 10 minutes. And if something has really never happened before, like my alien invasion example, what have we learned by applying the rule of three? Honestly- perform the experiment yourself. Wait for X time, then calculate 3/X. Do you now have an upper bound on the probability that an alien invasion will happen?
>Wait for X time, then calculate 3/X. Do you now have an upper bound on the probability that an alien invasion will happen?
Assuming we would observe an alien invasion and write about it, we have about 5000 years since humans started writing. So 3/5000a, so an upper bound on an alien invasion in a year (assuming p hasn't changed since we started writing) is 0,06% per year.
It's an upper bound in any case, meaning that is this or less. That includes p = 0. You're just 95% confident that it won't be higher than that.
But maybe the aliens brought us our writing system.
Re: Estimating the chances of something that hasn’t happened yet
#136Earlier quoted context omitted.
In the case of a book the first pages are likely to be significantly better than the rest. An editor knows that once you have invested in reading the first part of the book you are unlikely to put the book down. This means the first pages have to suck you in, and not do anything to make want to quit reading. The most important part is the first sentence as this is often the only part that drives your buy/leave in the…
> The most important part is the first sentence as this is often the only part that drives your buy/leave in the store decision. I usually sample books from the middle. I wonder what the heatmap of bookstore reading really looks like?
My informal sampling suggests most people start at first page. They worry that reading the middle or end would reveal a spoiler. It would be interesting to get good data though.
Re: Estimating the chances of something that hasn’t happened yet
#137Earlier quoted context omitted.
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.
You always have more data -background knowledge- unless you've only existed in those last 10 minutes. And if something has really never happened before, like my alien invasion example, what have we learned by applying the rule of three? Honestly- perform the experiment yourself. Wait for X time, then calculate 3/X. Do you now have an upper bound on the probability that an alien invasion will happen?
Yes, I do have a reasonable (95% confidence, as per article) upper bound on the probability - as I haven't noticed an alien invasion during my lifetime, it seems reasonable to conclude that noticeable alien invasions are very rare events and happen less frequently than once every decade. Possibly much less frequently, possibly many orders of magnitude less frequently, possibly never, but that's how upper bounds work.
Re: Estimating the chances of something that hasn’t happened yet
#138I'd be interested to know how to estimate things that are very rare or don't have a normal distribution. For instance, let's say I have a bold plan to protect us from meteor strikes that will cost $100B. How would a person make a decision about whether that's a good trade-off or not? And how would a mathematician help them make that decision? How would it change for more complex cases, like a shield to prevent nuclea…
Simple answer, this is a rough upper bound. The more information about your problem you have, the better you will be able to model the probability distribution and the better priors you have on it happening.
Re: Estimating the chances of something that hasn’t happened yet
#139I don't know the first thing about schools of thought in statistics (frequentist? Bayesian?) but something feels fishy about extrapolating a probability based on sample size (or number of trials) alone. 20 pages, no typos, Would someone care to explain why the math says it shouldn't matter (for reasonably small values of p)?
>per page
Re: Estimating the chances of something that hasn’t happened yet
#140This is a useful rule of thumb.
Lets try stretching it: Humans havent destroyed the world in their 300,000 years of existence, so probability of them destroying the planet in future is less than 0.00001 (1 in 100,000). I feel like that might be an underestimate.
Reminds me of the Doomsday Argument