#h2 math

[biohazard2017]

the academic stress really is kicking in but for the wrong things 😭

[fresver]

Crack the H2 Code: A Student's Guide to Math Tuition Triumphs

In the intricate and demanding terrain of H2 Math, a subject fraught with complexities and nuanced concepts, students frequently find themselves in search of reliable guidance to successfully unravel the intricacies of mathematical theories and emerge triumphant in their examinations. In response to this academic challenge, this comprehensive guide aims to provide students with an arsenal of practical strategies tailored specifically for navigating the demanding landscape of H2 Math tuition. As the intricacies of the subject pose unique hurdles, ranging from advanced calculus to abstract algebra, the need for a clear, systematic approach becomes increasingly apparent. In the upcoming sections, we will delve into these strategies, emphasising a pragmatic and results-oriented methodology designed to empower students to not only comprehend but master the intricate code of H2 Mathematics.

Understanding the H2 Math Landscape

Navigating H2 Math in Singapore necessitates a clear understanding of the academic terrain. H2 Math Tuition steps in as a key ally for students, providing targeted support to grapple with the intricacies of the syllabus and enhance comprehension.

Strategies for Success

1. Prioritise Core Concepts: Start by focusing on core mathematical concepts. H2 Math Tuition emphasises a strong foundation, ensuring students grasp fundamental principles before delving into complex topics.

2. Practice Regularly: Consistent practice is paramount in H2 Math. Tuition programs structure practice sessions to reinforce learning, helping students become adept at applying mathematical theories to problem-solving.

3. Target Weaknesses: Identify and target weaknesses systematically. H2 Math Tuition hones in on individual weaknesses, providing tailored exercises to fortify areas where students need improvement.

4. Embrace Problem-Solving Techniques: H2 Math is as much about problem-solving as it is about theory. Tuition equips students with proven problem-solving techniques, enabling them to tackle even the most challenging questions confidently.

Maximising the Tuition Experience

1. Active Engagement: Actively engage in H2 Math Tuition classes. Participating in discussions and exercises fosters a deeper understanding of concepts, ensuring each tuition session becomes a valuable learning opportunity.

2. Utilise Supplementary Resources: H2 Math Tuition often comes with supplementary resources. Make the most of these materials, including practice exams and additional study guides, to reinforce learning outside of tuition hours.

3. Seek Clarifications Promptly: Don’t let doubts linger. H2 Math Tuition provides a conducive environment for students to seek clarifications promptly, ensuring that misconceptions are addressed immediately.

4. Stay Consistent: Consistency is key in H2 Math Tuition. Regular attendance and consistent effort contribute significantly to a student’s ability to absorb complex mathematical theories and apply them effectively.

Excel in Examinations

1. Effective Time Management: H2 Maths Tuition emphasises the importance of time management. Through strategic planning and effective utilisation of study time, students can navigate the demands of the H2 Math examination efficiently.

2. Review and Revise Systematically: Systematic revision is a cornerstone of H2 Math Tuition. Regularly revisiting concepts and practising past papers are integral parts of preparing for examinations.

3. Simulate Exam Conditions: H2 Maths Tuition often incorporates mock exams. Simulating examination conditions helps students familiarise themselves with the format, manage time constraints, and build confidence for the actual assessment.

4. Track Progress: H2 Maths Tuition provides tools for tracking progress. Regular assessments and feedback enable students to monitor their advancements, identifying areas that require additional attention before the examination.

Conclusion: Triumphing in the H2 Math Maze

Cracking the H2 Code is no small feat, but with the right strategies and the support of H2 Math Tuition, students can triumph over the challenges that the subject presents. By prioritising core concepts, embracing consistent practice, and actively engaging in tuition, students can navigate the H2 Math landscape and emerge victorious in their examinations. H2 Math Tuition serves as an educational supplement and as a strategic partner in a student’s journey to conquer the intricate code of H2 Mathematics. Contact Axiom Education Singapore today to learn more.

[asyncmeow]

how shinigami eyes works

first - the short version:

shinigami eyes is a browser extension that uses a fancy list called a bloom filter to identify people who can be considered transphobic or trans-supportive. a bloom filter is a kind of list that lets you check if someone is in the list, without being able to actually know the contents of the list. i go below into how it works, but the short version is Math™️, with a side effect of having false positives - thinking an item is in the list when it really isn't. whenever a social media username is detected (the exact method how depends on the website), it checks the username against that list, and applies some CSS to change the color of the text depending on what list the username is in.

the long version

shinigami eyes can be split into a few parts:

bloom filters

submissions

name highlighting

bloom filters

i'll start with the most complex part - bloom filters. the most common misconception about Shinigami Eyes is this: the filters are not updated in real-time. they are shipped with the extension which had last been updated since november 2022, according to the FF extension site. in other words: nothing marked since then can be seen by anyone other than who marked it. you can see that in the code here, where it loads the bloom filters from a data/[something].dat file included in the extension (but not in the github repo).

the following information about bloom filters is my summarized version of this page.

bloom filters are, in a slightly longer explanation than before, a way to know if an item is *not* in a list with 100% certainty, but there's a false positive rate that grows as more names are added to the list. a bloom filter of a single size is able to handle any number of items in the list, though. there's also the issue that you can't *delete* stuff from a bloom filter - you would need to regenerate it from scratch to do that.

now that the medum-sized explanation of what they are is done, let's go into how they work. a bloom filter is a set of n bits, initially all set to zero. to add items into the filter, you need a few hash functions, in this example i'll use h1, h2, and h3, with n=10. if I wanted to add the text asyncmeow to the list, i would do this: h1("asyncmeow") % n // n = 10, h1(...) % 10 = 9 h2("asyncmeow") % n // n = 10, h2(...) % 10 = 5 h3("asyncmeow") % n // n = 10, h3(...) % 10 = 8

after that, i have a list that looks like this (keep in mind that the list is zero-indexed):

[0, 0, 0, 0, 0, 1, 0, 0, 1, 1]

you can then check if something is in the filter by running the same hashing functions and checking if the result bits are set in the filter. you can access the bloom filters used for shinigami eyes by going to about:debugging#/runtime/this-firefox in Firefox and clicking "Inspect" on Shinigami Eyes, then running bloomFilters in the console. as this could possibly change in an update, i don't want to go into how they are set up, and i haven't dug enough into how their bloom filter code works well enough to say anyways.

submissions

submissions on shinigami eyes are encrypted (as in - encryption separate from HTTPS), then posted to https://shini-api.xyz/submit-vote. you can see the code for this here. when you right click someone to mark them, their name is stored in the local data of your browser in an overrides property.

name highlighting

name highlighting is done by checking them against the bloom filters and your local overrides. if a user is present in either bloom filter, or present in your local overrides, they are marked accordingly. not much to it from there.

that's really it, i think? feel free to ask if you have any questions! nya :3

[conchetttecurls]

Rosewood Readathon Day 5!

Fixing your math grade is like entering a different reality. I have weeks of my life from like january to april 2022 blocked out because i just spent all my time doing maths. (ended up with a H2 though so hehehe go me). So I feel you Lottie

Bimdas anyone?

I remember being 12 and thinking this math equation was the most confusing thing in the world. Now though I feel like I could solve it if I put my mind to it. Or else I'm just gonna give it to my friend who's doing a maths degree and see will it knock his ego down a few pegs

I don't think I've mentioned this before but I love how Lottie is a huge fan of fairytales. She is like me frfr (except I'm a huge fan of Irish/Celtic mythology)

Break in time!!! What a fun little scene. However, it says they got completely soaked going through the river so how did they not get caught in the morning assuming they left a trail of water after them?

Also what a bunch of nerds going to the library on the orientation day. Never forget we are reading about pretigious, rich nepo baby nerds everyone. (I still love them all with my entire heart)

[white-flwrs]

14.11.2024

i crashed h3 bio lesson in school this morning, just to see how the subject is being taught and because a lot of my friends wanted to take it. i'm pretty sure i was the only physics-taker there. anyways im so glad i didnt take bio as it seems to truly be way too hard to score for. h2 bio itself has a 15 mark essay question and most people only get around 4-5! h3 bio is even better: the entire paper consists of 3 essays and that's all! at this point it should be considered a humanities subject. i had a h3 maths lesson later that day which went quite alright. i'm starting to warm up to the subject, and i love that some of my close friends are in my class and we get to sit together. i didn't study much myself today. i redid my notes for temperature & ideal gases as i wasnt the most clear with that topic, that's all.

earlier in the year i'd found a telegram bot which matches people with other people in our school just to make new friends. i matched with someone today and we were chatting and we really hit it off! got so much tea today from this guy, i didn't know half the drama that was going on with even the people i thought i knew well! we were quite engrossed in chatting that i only slept past 1, oops... but it's alright it was fun

xx

[adj444]

HELLO i'm in college now so you can access my leaving cert posts here, the grades i got for LC 2023 for your reference when reading the selection of notes i posted ARE:

H1 biology & english

O1 french & math

H2 ag. sci

H3 business (because i didn't finish the exam fml)

H4 art (overall, not just art history)

but because making my silly little notes and doing my silly little daily overviews helped me so much i'm going to continue doing it but for NURSING SCHOOL!!!!!! so if you continue to follow then you're going to be getting a lot of biology pt. 2 and nursing theory. THX 4 THE LC SUPPORT!!!! :)

[farukseo]

I will website on page optimization that google loves

Google ranks recent websites based on On-page optimization.

A website never ranks in Google without On-page optimization. The best Wordpress SEO plugins I would use Yoast or Rank math. Besides , I will do competitors and keyword Research through various tools. For example: SEMrush , Ahrefs and keyword Revealer.

However ,without wasting any more time set up these below tasks to increase your website. Or hire me,

Meta Title set up

Meta description set up

Image Alt text

URL Optimization

Keyword Research

H1 h2 H6 tag Set up

Internal links 

External links 

Google Analytics verify 

Schema markup set up

Google search console set up

Sitemaps set up 

Robot .txt setup 

Google Tag manager set up

404 & Redirection    

 All links index

[debbielearningcove]

Why H2 Math is Essential for Acing University-Level Courses in Singapore

In Singapore, the academic journey from junior college (JC) to university is a challenging path that demands rigorous preparation, especially in mathematics. One of the most critical subjects for students aspiring to pursue science, engineering, economics, or related fields at the university level is H2 Math. This subject is more than just another hurdle on the way to the A-levels; it is a foundational course that prepares students for the complex mathematical concepts they will encounter in their higher education. For many students, excelling in H2 Math is not just beneficial but essential for success in university-level courses.

Understanding H2 Math

H2 Math, short for Higher 2 Mathematics, is a subject offered in Singapore's junior colleges as part of the A-level curriculum. This course is designed to provide students with a solid foundation in mathematical concepts and techniques that are crucial for higher education in various fields. Unlike H1 Math, which offers a more basic understanding of mathematics, H2 Math delves deeper into advanced topics such as calculus, algebra, geometry, and statistics.

The curriculum is rigorous and comprehensive, covering a wide range of topics that are essential for students who plan to pursue university degrees in disciplines like engineering, computer science, physics, economics, and even certain areas of social sciences. The skills and knowledge gained in H2 Math are directly applicable to these fields, making it a critical component of a student's academic preparation.

The Importance of H2 Math for University Success

1. Foundation for Advanced Mathematical Concepts

University courses, especially those in STEM (Science, Technology, Engineering, and Mathematics) fields, often involve complex mathematical theories and applications. Subjects like engineering, physics, and economics require a strong understanding of calculus, differential equations, and linear algebra. H2 Math provides students with the foundational knowledge in these areas, ensuring that they are well-prepared to tackle the challenges of university-level coursework.

For instance, in engineering courses, students will encounter advanced calculus and differential equations that are crucial for understanding physical systems and solving real-world problems. Without a solid foundation in H2 Math, students may struggle to grasp these concepts, leading to difficulties in their university studies.

2. Critical Thinking and Problem-Solving Skills

H2 Math is not just about memorizing formulas and solving equations; it also emphasises critical thinking and problem-solving skills. The subject challenges students to think logically, analyse complex problems, and devise innovative solutions. These skills are essential for success in university courses, where students are often required to solve open-ended problems and think critically about complex issues.

In fields like economics, for example, students must use mathematical models to analyse economic data and make informed decisions. H2 Math equips them with the ability to approach these problems methodically, ensuring that they can apply mathematical principles effectively in their analysis.

3. Preparation for Competitive University Admissions

In Singapore, university admissions are highly competitive, especially for popular courses like engineering, computer science, and economics. Admissions committees often look for students who have demonstrated excellence in subjects that are relevant to their chosen field of study. A strong performance in H2 Math can significantly enhance a student's application, showcasing their readiness for the rigours of university-level courses.

Moreover, certain universities and courses may require or strongly recommend H2 Math as a prerequisite for admission. For students aiming to secure a place in these programs, excelling in H2 Math is not just an advantage but a necessity.

4. Application in Real-World Scenarios

Many university courses, particularly in the STEM fields, involve the application of mathematical concepts to real-world scenarios. Whether it's designing a bridge in civil engineering, developing algorithms in computer science, or analysing financial markets in economics, the ability to apply mathematical theories to practical problems is crucial.

H2 Math provides students with a strong foundation in applied mathematics, preparing them to use mathematical models and techniques in their university studies and future careers. This practical application of mathematics is what sets H2 Math apart and makes it an essential subject for students pursuing technical or analytical fields.

5. Bridging the Gap Between JC and University

The transition from JC to university can be challenging for many students, particularly in terms of academic expectations and workload. H2 Math plays a crucial role in bridging this gap by providing students with the skills and knowledge they need to succeed in their university courses.

The rigorous nature of H2 Math ensures that students are accustomed to the level of difficulty they will encounter in university. This familiarity with challenging material helps ease the transition and prepares students for the more complex and abstract concepts they will face in their higher education.

The Role of H2 Math Tuition in Achieving Success

Given the importance of H2 Math for university success, it’s no surprise that many students seek additional support through H2 Math tuition. While the JC curriculum is designed to prepare students for the A-levels, the challenging nature of H2 Math often requires extra guidance to fully master the material.

1. Personalised Learning Experience

One of the key benefits of H2 Math tuition is the personalised learning experience it offers. In a typical JC classroom, teachers must cater to a large group of students with varying levels of understanding. This can make it difficult for some students to keep up with the pace of the lessons or to receive the individual attention they need.

In contrast, H2 Math tuition provides a more tailored approach, with tutors focusing on each student's specific strengths and weaknesses. This personalised attention allows students to address their particular areas of difficulty, ensuring that they fully grasp the concepts and techniques required for success in H2 Math.

2. Targeted Exam Preparation

H2 Math tuition also plays a crucial role in preparing students for the A-level exams. Experienced tutors are familiar with the exam format and can provide targeted preparation to help students excel. This includes practising past exam papers, understanding the marking schemes, and developing effective exam strategies.

By focusing on the specific requirements of the A-level exams, H2 Math tuition helps students build the confidence and skills they need to perform well on test day. This, in turn, enhances their overall university application, as strong A-level results in H2 Math are often a key factor in admissions decisions.

3. Supplementary Resources and Support

H2 Math tuition often provides students with supplementary resources that are not available in the standard JC curriculum. These may include additional practice problems, study guides, and online resources that help reinforce the material covered in class.

Furthermore, H2 Math tutors can offer ongoing support and guidance, answering questions and providing clarification on challenging topics. This additional support can make a significant difference in a student's understanding and performance in H2 Math, ultimately leading to better results and a stronger foundation for university studies.

H2 Math is an essential subject for students in Singapore who aspire to excel in university-level courses, particularly in STEM fields and other disciplines that require strong mathematical skills. The foundational knowledge, critical thinking abilities, and problem-solving skills gained through H2 Math are invaluable for success in higher education and beyond.

For students who are serious about their academic and career goals, investing in H2 Math tuition can provide the extra support needed to master this challenging subject. At Debbie Learning Cove, we offer top-notch H2 Math tuition in Singapore, helping students build the confidence and competence they need to excel in their A-levels and prepare for university success. With our experienced tutors and personalized approach, we're committed to helping every student reach their full potential in H2 Math.

If you're looking to ace H2 Math and secure your place in a competitive university program, consider enrolling in our H2 Math tuition classes today. Let Debbie Learning Mathematics guide you on the path to academic excellence and a bright future

[fifthyeardiaries]

01/12 Check-In

Featuring a picture from my trip to Galway 2 weeks ago bc I'm hopeless at taking aesthetic pictures (my friend took this one).

What I did:

Studied Maths (1hr) - I have a test next Thursday and I want to do well because my last on didn't go so well and I feel bad because my teacher said I have at least H2 (80%+) potential

Blocked out time to study over the next 2 weeks - I have 4 tests coming up (2 next week and 2 the week after)

📖 currently reading: Solitaire by Alice Oseman

[seoexpert-molliksazzadurrahman]

Increase Organic Web Traffic — The Role of an SEO Specialist in Italy

Competitive landscape of digital marketing, the need for an SEO specialist has become more pronounced than ever. As businesses effort to increase their online visibility and effectively reach their target audience, the role of an SEO specialist in Italy has gained significant importance. In this article, we will explore key aspects related to SEO specialists, their costs, global recognition, demand, and the best SEO expert in Italy — Mollik Sazzadur Rahman.

Increase Your Brand Name — Boost Up Your Local Business Doing Local SEO for local business alone will not get the right results. At present you need to increase your Brand Value. SEO is not everything to get traffic. Yes it is true SEO is good for getting organic traffic. It’s better that you grow your brand name (by social media marketing or physically marketing) and I handle your SEO part.

How much does an SEO specialist cost?

Investing in the services of an SEO specialist is a strategic move for businesses aiming to thrive in the online sphere. The cost of hiring an SEO specialist can vary based on several factors, including the scope of work, the complexity of the project, and the level of expertise required. On average, businesses in Italy can expect to pay anywhere from €500 to €3000 per month for professional SEO services. It’s crucial to view this as an investment rather than an expense, considering the long-term benefits and increased visibility an SEO specialist can bring to your business.

Who is The World’s No. 1 SEO Expert?

Identifying the world’s No. 1 SEO expert is a subjective matter, as the field is vast and constantly evolving. However, there are several renowned figures who have made significant contributions to the SEO industry. Recognized names include Rand Fishkin, Neil Patel, and Brian Dean. These experts have demonstrated their prowess through innovative strategies and a deep understanding of search engine algorithms. While ranking the world’s №1 SEO expert may be challenging, it’s essential to acknowledge and learn from the thought leaders shaping the industry.

And among them, the name of Mollik Sazzadur Rahman has to be mentioned as an SEO expert. Even if he is not the world number one, but he can improve any local business by SEO services.

Mollik’s SEO Skills As an Optimizer for Website As Website SEO expert Mollik can do everything. But social media marketing is not his job. So,

Profitable Keyword Research

Content Analysis

On Page SEO

Used LSI Keywords into H2-H6 Tag

Image Optimization

Yoast SEO, Rank Math specialist

URL, SEO Title & Meta Description Utilize

Schema and Canonical Tag Generate

Technical SEO

Sitemap, Robots.txt Setup

Redirection, and 404 error solved

Page Speed Optimization

Off-Page SEO

Guest Post, Web 2.0

Social Bookmarking

WordPress, Shopify and Wix Management

Google Analytics specialist

Google Search Console Analysis

SEO Audit Services

Competitors Analysis & Information Spy

Weekly & Monthly Report

Are SEO experts in demand?

The demand for SEO specialists continues to surge as businesses recognize the pivotal role search engine optimization plays in their online success. In Italy, as elsewhere, companies are seeking skilled professionals to optimize their websites, improve search engine rankings, and drive organic traffic. With the digital landscape becoming increasingly competitive, businesses are realizing the need for specialized expertise to navigate the complexities of SEO. This growing demand presents a promising opportunity for individuals looking to pursue a career as an SEO specialist in Italy.

In conclusion, the role of an SEO specialist in Italy is pivotal in the digital age, and the demand for skilled professionals in this field is on the rise. Businesses seeking to stay competitive and enhance their online visibility can benefit significantly from the expertise of an SEO specialist. Mollik Sazzadur Rahman, with his experienced skilled record and commitment to excellence, stands as a leading SEO expert in Italy, ready to guide businesses towards online success.

[digitalexpertrana22]

I will do onpage SEO and technical optimization for website ranking

Hey Dear,

Are you eager to see your website achieve a top ranking in Google search engine results?

Don't worry, I specialize in enhancing your website's performance and ranking on Google through effective On-page SEO and Technical SEO techniques.

I will be utilizing WordPress plug-ins such as Yoast and Rank Math with my experience, and I will implement comprehensive On-page SE0 optimization in accordance with search engine policies.

On-page SEO Optimization Included:

SEO Title

SE0 Meta Description

Keyword Optimization

URL/Permalink Optimization

Heading Tags (H1, H2, H3, etc,) Setup

Keyword Density Check

Image optimization And Alt Tag

Internal/External Linking

Add Related Keypharse

Premium Yoast/Rank Math Plugging Setup

SE0 Audit Report

Technical SE0 Optimization Included:

Google Search Console

Website Speed

XML Sitemap

Robots.txt File

Canonical Tag

404 Error Pages

301 Redricting

Schema Markup

Benefits Of Choosing Me?

Ethical SE0 Practices

After-Purchase Service and Support

Ensuring 100% Client Satisfaction

Note: Kindly contact me before making the purchase to discuss your specific requirements.

Cheers, MD Rana

Thanks

[soikat3212]

I will do deep complete onpage seo for google ranking

If you are looking for a high-quality On Page SEO Service that can rank your site high on Google SERPs with pure white hat Onsite and Technical SEO Optimization then you are at the right place.

Improve your website rankings with proven On-Page and Technical SEO techniques for WordPress with Rank math or Yoast Plugin.

Why me?

4 Years of Experience in the SEO industry

100% White Hat Work

100% Customer Satisfaction

24/7 support

Long term relationship

On-Page Optimization

- Yoast/Rank math Plugin Setup

- Keyword Research

- Title Tag- Meta Description

- H1, H2, H3 Tags

- Internal Linking

- Permalink Structure

- Proper ALT Text for Images

- Broken Link Issues

- Connect with Google Analytics & Google Search Console

Technical Optimization

- Setup XML Sitemap

- Canonical URL

- Setup Robots.txt

- Web Pages Crawling & Fetching

- 404 & Redirection Issues

- SSL Certificate (For HTTPS Secure website)

- Setup Schema Markups (Product Schema, Local Business, Organization, etc)

Still have questions? Check the FAQ or send me a message.

Lets get started!

[max1461]

Ok, I have a scenario I want to model probabilistically, and ideally do hypothesis testing and so forth on, but I'm not exactly sure how to do it. If anyone could point me to a source which describes how to solve this general sort of problem (or even just give me terms to google), that would be very helpful. I want to be able to solve problems like this generally, not just this specific one. Ok, here goes:

Ok, so basically... I have a hypothesis H, which makes predictions about a data set X. In particular, it makes predictions about two related properties of the data, call those properties P1 and P2. So for any x in X, we might have P1(x) or P2(x) or both or neither, and H makes predictions about which x these properties should hold for. Let's refer to these predictions H1 and H2, respectively. So H1 is trying to predict P1 and H2 is trying to predict P2. There are certain other practical constraints placed on what H1 and H2 can be; they aren't allowed to be arbitrary truth functions but must satisfy additional requirements.

Ok, all good so far. It's easy to do hypothesis testing on H2, because it's easy to calculate what the frequency of P2 should be in the data by pure chance. If H2 does sufficiently better than chance at predicting P2, we can reject this null hypothesis, etc. H1 is a little harder because we don't have a clear prior for when P1 should hold by chance, but don't worry about that yet.

The next part is where I start to get fuzzy on the math. There are basically two aspects of this that I don't know how to handle.

The first part is this: we don't actually have H, we are trying to work backwards to determine whether a sufficient H exists. It's possible that the data are related by nothing but chance. The real hypothesis H' that we are interested in is "a hypothesis H which meets [practical constraints] and which does sufficiently better than chance at predicting the data in fact exists" and the null hypothesis we are trying to reject is "there is no hypothesis H which meets [practical constraints] and which does sufficiently better than chance at predicting the data". So, ok, I am not sure how to go from individual hypothesis testing to testing for the existence of hypotheses.

In the general case my intuition is that this isn't doable, "hypothesis space" is just so big that you can always find one that's better than chance. But in my situation, [practical requirements] are highly restrictive, so in fact the mere existence of a much-better-than-chance model meeting these requirements is surprising. It's actually this that I want to hypothesis test on.

Ok, that's part one of what I don't know how to do. Here's the second, even bigger thing. The two properties P1 and P2 are related to each other in a very specific way. In particular, if P1(x) holds for a given x and the hypothesis H is true, we should expect P2(x) to hold with greater probability. If P2(x) holds for a given x and the hypothesis H is true, we should expect P1(x) to hold with greater probability. If the hypothesis H is not true, we should expect P1 and P2 to be basically unrelated to each other.

In fact, it may not just be P2 which is related to P1 in this way. In the real situation, there is actually a whole set of properties P2,...,Pn that we are trying to predict, and they're all related to P1 like this. So if H is true and P1(x) holds, we should expect to see a greater proportion of P2(x), ..., Pn(x) holding. And the more of P2(x), ..., Pn(x) hold, given H, the more likely it should be that P1(x) holds. If H is false, there should be no relationship. And so, intuitively, we are looking for a collection H2,..., Hn that are heavily "inter-reinforcing", above and beyond what is expected by chance. If H_i and H_j are both true of x for a lot of the same x, then H_i and H_j are sort of providing evidence for each other. If a big collection H2,..., Hn exists that all provide evidence for each other, that is strong evidence that a hypothesis H satisfying our requirements exists. If no H2,..., Hn can be found, we fail to reject the hypothesis that P2, ..., Pn are related by nothing but chance.

This is why I said the fact that we can't really hypothesis test on P1 shouldn't matter, intuitively it feels like we can just ignore it and focus on P2, ..., Pn. But I'm not sure.

Anyway, that's basically the thing I'm trying to figure out. It is possible that I have either misdescribed this scenario slightly, or that there are other relevant details that I have failed to consider. That's why I'm hoping for a resource or general starting point for solving problems like this, because the exact nature of the problem may evolve over time. If you've got any thoughts, let me know!

[crazy-pages]

I have good news for everyone who's existentially freaked out by the concept of atomic emptiness.

"Everything is empty" is actually only true if you believe in a century old model of particles as tiny point masses. The reality once you start playing with the quantum math is that electrons are very large fuzzy clouds of like ... well, clouds tbh. They're clouds. Not as in, there is a chance of finding them in a specific position with this probability, or this position with a different probability. The particle itself is a cloud which simultaneously exists in all those places. Denser in some places than others, and in various weird shapes around the atomic nuclei. But these clouds are huge and take up a lot of space and they're constantly overlapping with other electron clouds.

(But then what's happening when you measure a specific location and a particle shows up there or doesn't with some probability, you might ask? That's for another post, I don't have time to explain bra operators to y'all.)

There's actually a ton of physical overlap between particles. Basically none of the space inside solid matter is unoccupied. As an example, the red line is what a cross section of the electron cloud densities of H2, diatomic hydrogen, looks like. (ignore the blue line). The hydrogen nuclei end up at those peaks you see. Now more complicated molecules get a lot more complicated than this, but it gives you an idea of the size scale of electron clouds are and how they can be spread out.

Well in solid matter like iron or silicon or what have you, a good rule of thumb is that atoms tend to be separated by 100~300 pm. And when atoms are connected like that, the electron cloud between them tends to look more like the center of the red line than the edges. So yeah, there's actually a ton of electron in the "empty space" between nuclei, and it's not all that much less dense than it is near the nuclei. Especially cus outer layer electron orbitals tend to be a lot bigger than what you see with hydrogen.

So yeah. The space between atoms is filled with stuff.

“nothing is real atoms never touch each other youve never touched anything in your life” ok. well when i pet my dog he is soft and when he licks my hand it is wet and that is far more real to me than whatevers going on at an atomic level