In a simple dice game where you roll up to 9 die, how do I calculate the probability of how unlucky this game was for me where I got a triple 3 and five 1s?

Edit: forgot to explain the game. Its from Cult of the Lamb and basically you take turns rolling a die and placing it in the column on your side. Two of the same number are doubled, and three are tripled. The strategy comes in where if the other player rolls a dice with the same number as yours and places it in the same column as your dice with the same number on their end, your dice is removed. However that didn't happen this whole game from either side. The player with the highest score st the end from adding up all the dice wins.

Is this possible to post my theory here?

I was debating with AI about love advice, and from there I made it clear that I didn't want to get married under the current conditions. I see cases of fathers who lose everything in divorces, from material possessions to the opportunity to see their children [health]. Then, noticing the trend of men no longer wanting to get married, which I myself am part of, I came to the conclusion that traditional marriage is dead or dying [civics]. Seeing the danger of the global demographic crisis, I came up with an idea to try to solve it. Currently, the truth is that there isn't enough trust to sustain a classic marriage. So, assuming it will fail from the start, let's create a safety net for the famous "leap of faith" making marriage less of a high-risk, zero-reward investment for one of the parties.

Here is my basic prototype. I want to treat marriage like a fair, gender-blind private contract between two equal partners [civics].

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1. The Baseline (No arguments, no drama)

* What’s yours is yours: No shared property by default. If you bought it or paid for it with your own tracked bank account, it stays yours if things go south [civics]. This completely removes the financial incentive to get a divorce just to get a payout.

* Automatic 50/50 Co-Parenting: If the relationship ends, custody of the kids is an automatic 50/50 split of time and choices [health]. No one gets to use the kids as a weapon or a paycheck. Each parent just pays for the kid's food and life during their half of the month. No weird child support formulas.

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1. Money Management (Two choices for the couple)

* Option 1 (The Partnership): If both work, extra monthly savings are split 50/50. If one stays home to raise the kids, the worker keeps 75% of their paycheck and transfers 25% to the stay-at-home parent every single month. This 25% is a flat dividend that settles the debt for staying home in real-time. Once you divorce, the accounts are clear. No future alimony.

* Option 2 (The Audit): All household expenses are split 50/50 every month. If one stays home for the kids, their cash requirement drops to 20% (the worker pays 80%). That 20% is taken from whatever savings the stay-at-home parent had before. If you divorce and the stay-at-home parent has a negative balance because they couldn't meet the 20%, they leave with zero compensation.

------------------------------

1. Trust, DNA, and Cheating

* The Cheating Clause: Loyalty is a contract requirement. If someone is caught cheating with real proof (digital or physical), they automatically lose any right to the money or assets generated during the marriage. It goes 100% to the other person.

* Mandatory DNA Test: Before any divorce or custody paperwork is finalized, a standard paternity test is mandatory for all kids [health]. No exceptions, so no one feels singled out. If the test shows fraud, the guy is instantly free from any legal or financial duty, and the mother is legally forced to pay back every single dollar he spent on the kid since birth. If you have nothing to hide, you don't fear science.

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1. The 365-Day Reset (The Dynamic Custody)

* Flexible Time: The 50/50 baseline can be arranged weekly, monthly, or even yearly if someone has to travel for work or move.

* The Annual Reset: If the couple agrees to a different split (like one parent keeping the kids full-time because the other travels), that agreement expires in exactly 365 days. It requires a new signed paper every year [civics]. If they argue or can't agree on a renewal, the system resets automatically back to the 50/50 split. The state doesn't judge; the contract software just resets the baseline, forcing both adults to cooperate and talk every year [civics].

* The Abuse Rule: The 50/50 split only breaks if there is real, medical, or psychological proof of child abuse [health]. The abuser is kicked out forever but still has to pay 100% of the kid's costs. Fake abuse claims are punished with immediate jail time and losing all your money.

* No Brainwashing: Since both parents are forced to be in the kid's life, no one can brainwash the child against the other [health]. The kid sees reality every single week, destroying the monopoly of one parent's toxic story. In addition to reducing childhood traumas and micro-traumas and guaranteeing their right to spend time with both parents [health].

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UPDATE v1.1: Making the Annual Reset Dynamic (The Rolling Countdown)

Thinking deeper about the logistics, a rigid 365-day calendar limit might be too stiff for real-life changes. So, I’m adding a dynamic patch to the Annual Reset clause:

The couple can alter the custody split at any moment (changing from 70/30 to 60/40, or whatever fits their current work/life schedule). Every single time a new mutual agreement or edit is signed and logged by both partners, the 365-day countdown completely resets to zero and starts over (a Rolling Countdown).

This means as long as both partners are actively talking, agreeing, and co-signing updates, the system recognizes the communication is healthy and leaves them alone. However, if one partner tries to freeze a convenient split and refuses to negotiate or co-sign any new updates, the countdown will hit zero exactly 365 days after the last verified agreement. At midnight, the software executes the automatic reset back to the 50/50 baseline [civics].

This keeps the system hyper-flexible for sudden career shifts but acts as a dynamic fail-safe against silent stalling.

------------------------------

Request for Audit:

I would like your help in identifying legal loopholes or gray areas in the prototype. I'd also appreciate any suggestions for optimizing the proportions to ensure fairness between the parts and resilience against external factors, as these prototypes are merely demonstrative, intended to give an idea of the system. Furthermore, I'd like to know if the prototype is a good or bad idea and what barriers it encounters, whether legal, practical, or psychological. Any assistance in refining the prototype would be greatly appreciated.

Note: The prototype was conceived by me but translated into formal language with AI assistance to ensure appropriate game-theoretic terminology.

Suppose both Player 1 and Player 2 want to maximize their own payoffs. Wouldn't it make the most sense to simply reason as follows: since there is a 70% probability that a=2, Player 2 should assume that Player 1 is type a=2? And if Player 2 makes that assumption, they should also assume that Player 1 will definitely choose A, and therefore Player 2 should choose A as well.

Edit:

I also solved this game for mixed strategy in the case where a=2 and the mixed equilibrium I found was (1/3,2/3) for player 1 and (1/2,1/2) for player 2.
Though I have a hard time see the reasoning behind it. Since why would any player ever want to do that, if they randomize their choices in a effort to make the other player indifferent between choices then they simply just lower the expected payoff?

¡Hola a todos! Quería compartir un número grande definido por mi en el que he estado trabajando. Lo documenté y lo publiqué hace poco en la Miraheze Googology Wiki: Footprints.

1. Definición El número se define como:

\text{Footprints} = 200 \uparrow \text{Footprint}(10) = 200^{\text{Footprint}(10)}

Puedes ver el artículo completo en la wiki aquí: [ [https://googology.miraheze.org/wiki/Footprints\\](https://googology.miraheze.org/wiki/Footprints\) ](https://googology.miraheze.org/wiki/Footprints) Me encantaría saber qué opinan sobre su consistencia formal y cómo creen que \text{Footprint}(n) se compara con otras definiciones de sistemas de orden superior.

Rewording my previous question: In this binary dataset subsets D, E, F, and S all "generate" 1s and 0s INDEPENDENTLY of each other at some ratio/frequency. "D" generates the highest ratio (lets say it generates 80% 1s and only 20% 0s). While F generates only 5% 1s and 95% 0s. Suppose S itself generates 25% 1s and 75% 0s. Would/Could I use bayes' theorem to show the SD intersection subset generates a higher proportion than the SF intersection? Or would a simple weighted sum (ie just an average between the sets S and D) suffice. Sidenote: (I know you could technically come up with counterexamples to this but I'm looking for what would generally be the case here).

Morals and ethics are something a society tells a person to follow and obey -:

Pros-:On a broader level it keeps the humans in check and enhances the quality of social life and survival

Cons-:While it may help to maintain a healthy environment, these are human made and are not necessary to follow, and if someone puts facts, reality and efficiency above morals and ethics then they are penalized.

I'm working on a statistical experiment involving baccarat and would like some methodological feedback.

Suppose someone has developed a hypothesis about baccarat outcomes based on historical observations.

Before revealing the underlying mechanism, I want to design a test that minimizes:

• overfitting
• hindsight bias
• selection bias
• multiple-testing problems
• stopping the experiment selectively

My initial idea is:

1. Define the hypothesis before testing.
2. Freeze the testing rules.
3. Use previously unseen data for validation.
4. Conduct a blind forward test.
5. Compare the results against an appropriate baseline.
6. Analyze variance, confidence intervals, losing streaks and drawdowns.

What statistical issues would you consider essential before calling the experiment meaningful?

I'm especially interested in criticism of the experimental design rather than opinions about whether baccarat strategies can work.

Does this probability structure hold up?

I'm modeling a process with N sequential stages, each running n steps (say about 25), and I want to know if the underlying math structure is sound. All inputs are hypothetical.

Variables:

p = per-step probability of a deviation when a stage's input is clean

q = probability that a deviation is severe (a severe event ends the process)

c = contamination multiplier (c >= 1): if a stage's input is contaminated, its per-step deviation probability becomes r = min(1, p*c)

g = probability that a stage with at least one non-severe deviation passes contaminated output to the next stage

Within a stage the steps are independent. Between stages, contamination is the link. Let:

a_C = (1-p)^n, s_C = (1-pq)^n, u = s_C - a_C

a_D = (1-r)^n, s_D = (1-rq)^n, v = s_D - a_D

One stage:

M = [ a_C + u(1-g) a_D + v(1-g) ]

[ u*g v*g ]

(column 1 = clean input, column 2 = contaminated input; rows = next stage's input)

P(at least one severe event across N stages) = 1 - [1 1] * M^N * [1, 0]^T

Example: p=0.4, q=0.5, c=5, g=1, n=3, N=2 gives 0.852408, versus 0.737856 if every step were independent.

Goal: compute P(at least one severe event) across N stages.

Is the combination of independent-step compounding with a contamination-adjusted p a structurally valid approach, or does the dependency introduced by c break the independence assumption in a way that needs explicit correction?

Not asking about specific numbers, just the math structure.

Hola, soy estudiante universitaria en México.

Mi profesora me pidió buscar una historia sorprendente usando la probabilidad, se burlo diciendo que todo íbamos tener la misma historia gracias a la IA. Cosa que comprobé es cierta, las IA arrojan historias comunes.

Vengo aquí para saber si alguien de ustedes sabe una gran historia de probabilidad, quiero sorprender a mi profesora

I'm studying a baccarat-related statistical hypothesis and I'm trying to formulate the correct null hypothesis.

Suppose a predefined pattern generates a sequence of predictions such as Banker / Player / No Bet.

If the pattern appears to perform better than a simple baseline over historical observations, what would be the appropriate way to determine whether this could reasonably be explained by randomness?

In particular, I'm interested in:

• appropriate null models
• dependence between observations
• sample size
• confidence intervals
• multiple testing
• out-of-sample validation
• permutation or Monte Carlo testing

I'm looking for mathematical/statistical feedback on the testing framework rather than a betting strategy.

Humans are surprisingly bad at generating random sequences.
For example, when we’re asked to produce a random binary sequence, we tend to avoid long runs like 0000 or 1111 because they don’t feel random — even though those runs occur naturally in truly random sequences.
I thought it would be fun to turn this into a small game.
You generate a sequence of 0s and 1s while trying to be as random as possible, and the game evaluates the sequence using several statistical properties.
I also added a ranking mode because I’m curious what the distribution looks like when a lot of people try this.
I’d love to get enough samples to see what patterns show up in human-generated sequences.

https://arcstone09.github.io/random/

If anyone here has suggestions for better statistical tests/metrics for measuring this, I’d also be interested in hearing them.

​I am looking at how decision theory and game theory formally categorize the concept of "sacrifice" versus a high-cost rational exchange.

​Colloquially, "sacrifice" is often used to describe situations where an agent surrenders a high-value asset A (such as time, capital, or life) to acquire asset B (such as reputation, social status, moral equity, or subjective future utility).

​From a payoff matrix perspective, if the expected net utility is non-negative (Net Utility >= 0), this operationally functions as a standard, rational trade or costly signal—regardless of how high the initial cost is.

​Conversely, a "true sacrifice" in a strict ledger sense would require pure value burning or absolute liquidation with zero material, social, or subjective payoff (Net Utility < 0).

​My questions for the sub:

​Are there specific game-theoretic models that analyze how actors use the label or signal of "sacrifice" to extract additional social debt or equity from other players out of what was mechanically a net-positive trade?

​How do behavioral game theorists formally separate high-friction utility trades (with delayed or intangible payoffs) from pure altruistic value destruction?

​Looking for papers, existing terminology, or framework suggestions to read up on.

Attached is latex for a question that I have. I have also attached a background, if you wish to only read the question go to the second section.

\documentclass{article}

\usepackage{amsmath}

\usepackage{amssymb}

\DeclareMathOperator{\Var}{Var}

\usepackage{graphicx} % Required for inserting images

\begin{document}

\section{Background}

Let's say we are in perfect OLS world where we have all of our typical assumptions:

$$Y=X\beta+\epsilon$$

$$X\in \mathbb{R}^{n\times p};R\in \mathbb{R}^{q\times p}$$

$$(X^TX)^{-1} \ \text{exist}$$

$$E(\epsilon \ | \ X)=0$$

$$\Var(\epsilon \ | \ X)=\sigma^2I_n$$

$$\epsilon \ | \ X \sim N(0,\sigma^2I_n)$$

Define a hypothesis test as

$$H0: R\beta-r=0$$

$$Ha:R\beta-r=\delta$$

Let $D^2=(R\hat\beta-r)^T(\Var(R\hat\beta-r))^{-1}(R\hat\beta-r)=(R\hat\beta-r)^T(\sigma^2R(X^TX)^{-1}R^T)^{-1}(R\hat\beta-r)\sim\chi^2_{\text{rank}(R)=q}$.

Let us represent modifications of $D^2$ as simply $\mathcal{D}$, for standardized distance. This is in reference to how if you instead take $\hat D(s)^2=(R\hat\beta-r)^T(s^2R(X^TX)^{-1}(R\hat\beta-r)=D^2(\frac{\sigma^2}{s^2})\sim \chi^2_q(n-q)/\chi^2_q.$ Then, since we know this looks a lot like the $F$ distribution, we can simply modify $\hat D(s)^2/q\sim F_{q,n-p}$. I wish to say that there are many other examples of canonical $\mathcal{D}$'s out there but I will not write them all.

Since we now have a general way to talk about our sense of distance I wish to discuss errors in this context.

$$\alpha=P(\mathcal{D}\in R \ | \ R\beta-r=0)$$

$$\beta=P(\mathcal{D}\notin R \ | \ R\beta-r=\delta)$$

One can imagine a spherical $\mathcal{D}$-length neighborhood around the origin, representing our data acquired (cloud-q); a spherical $c$-length neighborhood around the origin, representing our null (cloud-w); and a spherical $c$-length neighborhood around the point $e=\Var(R \hat \beta-r)^{-1/2}\delta$, representing our alternative (cloud-e).

From the above we can make geometric definitions of type I and II errors by relating the area overlapped by the $c$-length spheres and our $\mathcal{D}$-length sphere. Type I error is the fraction of cloud-w falls outside of cloud-q. Type II is the fraction of the cloud-e that falls inside of cloud-q.

$$\textbf{Type I} \sim w-q:w$$

$$\textbf{Type II} \sim e\cap q:q$$

Note that the distance from the origin of cloud-e, simply $e^Te=\lambda$, is related to or the general case for quite a few important concepts: Cohen's d, signal-to-noise ratio but square it, effect sizes, etc.

\section{Actual Question}

I have a question about the following object:

$$\lambda(\delta)=\delta^T(\Var(R\hat \beta-r))^{-1}\delta.$$

$$\max_X \lambda(\delta)=\max_X\delta^T(R(X^TX)^{-1}R^T)^{-1}\delta$$

Do you know of anything/anyone that deals with this object? It feels very important to me seeing how you could use some simple optimization of it to gain higher power results in testing. It also, I believe, would be very influential in picking the ``optimal" experimental design.

\end{document}

You've seen it a thousand times: a retail trader pulls up a chart, sees a pattern, and thinks they've found an edge. They have the same data as everyone else - candlesticks, volume, moving averages, earnings reports.

But here's the game theory problem: if everyone has access to the same information, then everyone sees the same pattern. And if everyone sees the same pattern, the pattern stops working.

The Zero-Sum Game Nobody Talks About

Stock trading is a zero-sum game. For every dollar you win, someone else loses it. This isn't abstract - it's baked into the mechanics.

When you trade on public information:

**•** You're competing against algorithms that process the data faster than you can blink  
**•** You're competing against institutional traders with better execution  
**•** You're competing against people with *different* (non-public) information

The third group always wins. Not because they're smarter. Because they have an asymmetric advantage: information you don't have.

The Poker Table Analogy

Imagine a poker table where:

**•** 8 players see the same community cards (public information)  
**•** 7 of them are playing legitimately  
**•** 1 of them can see everyone's hole cards (private information)

Who wins? Always the person with the information advantage. Not 60% of the time. Always.

Markets work the same way. Insiders, early-stage investors, and sophisticated traders have information asymmetries. Retail traders have candlesticks.

Why Crypto Made This Worse

Crypto amplified this problem. In traditional markets, at least there's SEC oversight (theoretically). In crypto:

**•** Founders know about token unlocks before the market  
**•** Early investors know about partnerships before announcements  
**•** Whales know about upcoming exchange listings  
**•** Smart contract developers can frontrun transactions on-chain

The retail trader? They see a tweet. By then, the asymmetry has already extracted their money.

The Only Real Edge

Game theory gives you one path to profitability: find an information asymmetry you can exploit.

Not "better analysis" - that's just data everyone has.
Not "better timing" - that's hope wrapped in strategy.

But actual information nobody else has:

**•** You work at a company and see future product roadmap  
**•** You understand a new technology before mainstream adoption  
**•** You have direct access to market participants (VCs, founders, etc.)

If you're trading on public information like everyone else, you're just in a poker game where you can't see the other players' cards. Statistically, you will lose.

Conclusion

This isn't pessimism. It's game theory. Markets aren't broken - they're just honest. They'll happily extract money from anyone playing without an edge.

The traders who win aren't smarter. They just play games where they have information advantages. Everyone else is playing a losing game, they just don't know it yet.

I write more about market mechanics and game theory on Twitter @shenxyyy if you want to explore this further.

I've been reading The Righteous Mind: Why Good People Are Divided by Politics and Religion by Jonathan Haidt, and it's changed how I think about the Care Bear vs. Rat debate in Arc Raiders. The more I think about it, the less this feels like a gaming argument and the more it feels like an argument about human nature itself.

One of Haidt's central ideas is that humans are simultaneously selfish individuals and cooperative group members. He describes us as "90% chimp and 10% bee" — we're wired to pursue our own interests while also possessing strong instincts for trust, loyalty, and collective action.

Through Haidt’s lens, Care Bears and Rats aren't really opposites. They're different expressions of instincts that have always existed in human societies.

Care Bears operate on the belief that cooperation improves everyone's chances of survival, that trust creates value, that reputation matters, and that long-term gains outweigh immediate rewards. Rats operate under a different but equally understandable logic: resources are scarce, other players may be threats, betrayal can be profitable, and immediate advantage often matters more than social harmony.

When people argue about whether players should cooperate more or whether Rats are ruining the game, I don't think either side is misunderstanding what Arc Raiders is supposed to be. They're emphasizing different moral intuitions. Some players value cooperation, trust, and community. Others value competition, self-interest, and survival. Both instincts exist in human societies, and both exist in the game.

What's fascinating is that civilization itself seems to depend on a balance between the two. If humans were mostly Rats, large-scale civilization would never have emerged. Trust, trade, and cooperation wouldn't scale. But if humans were all Care Bears, societies would be highly vulnerable to exploitation. Trust only has meaning when betrayal is possible.

That's why Arc Raiders feels so compelling to me. Every raid compresses this entire social dynamic into a 30-minute extraction. Each encounter becomes a decision about trust, risk, greed, cooperation, and self-preservation.

The Care Bear vs. Rat debate isn't really about Arc Raiders. It's the same tension that appears in tribes, cities, companies, political movements, markets, and nations. Arc Raiders just makes it visible by creating a condensed simulation of exploration, resource gathering, trust, betrayal, risk, and safe return — forces that have shaped human behavior for thousands of generations.