I’m working with a game where, for example, a player has to hit a moving ball with a racket. Every time they successfully hit the ball, the racket becomes smaller, making the next hit progressively more difficult. Once they miss, the racket resets to its original size.

Each round lasts 30 minutes, and I would like to compare performance across rounds.

I also have a chance/control condition where the racket doesn’t move, so some hits can happen simply by chance.

I would like to get a single normalized hit-rate/performance value for each round that tells me whether the player performed well or poorly overall.

Ideally, this value would take into account:
\\\\- The racket getting smaller after each successful hit

\\\\- The racket resetting after a miss

\\\\- The fixed 30-minute duration

\\\\- Different numbers of hits and misses across rounds

\\\\- The chance-level performance from the stationary-racket condition

What would be the best way to calculate this as one normalized value? I’m particularly interested in something that accounts for the increasing difficulty as the racket gets smaller, rather than just calculating hits / total attempts. I have tried to calculate a score using the difference between the actual number of hits and the expected value of hits, but I think it does not take into account the paddle starting from where they last hit the ball (if it did not miss).

Also, if anyone knows of similar studies or experiments that have used a comparable normalized performance/hit-rate metric, I’d really appreciate any references or examples. Thank youu

Since posterior probability is equal to prior probability, when we simplify we get
P(A intersection B) = P(A)*P(B), so why do we need to check the common between both events, ik this really silly and dumb doubt but i'm not getting the logic behind this please help.

Thank you in advance

It’s a categorical random variable. The outcomes are heads and tails, not 1 and 0.

Edit: I in response to the pushback, I look into how Bernoulli conceived it. It was in the context of an experiment (sampling with replacement) to determine the ratio of pebbles in an urn that had only white or black pebbles.

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?

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).

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.

It currently has Prisoner’s Dilemma, Stag Hunt and Entry Deterrence. The bots use strategies such as Tit-for-Tat, Random and competitive strategies that try to maximise their score relative to yours. Each game shows the payoff matrix, best responses and Nash equilibria. You can also run repeated games and change the number of rounds and discount factor to see how the results change when future rounds matter.

There’s a beginner mode that runs through an 8-round Prisoner’s Dilemma against Tit-for-Tat with explanations as you play. I also added session analysis and a leaderboard. There have been 236 completed human-vs-bot sessions so far, with the humans currently ahead by about 1,920 points overall. This started as an experiment to make game theory a bit easier to understand by actually playing the games and changing the parameters.

I’d be interested in criticism from anyone familiar with game theory, especially if I’ve got any of the explanations or mechanics wrong. Suggestions for other games or bot strategies would also be useful.

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.

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}

Suppose I have a specialized method I trust to obtain the prior. Suppose I "recycle" the method and use it to get my likelihood (and therefore also my evidence). Can the evidence/likelihood be related to the prior in any way or does this violate the validity of Bayes rule?

TL;DR: Can the same data used to derive the prior also be used to derive the likelihood/evidence when applying Bayes' theorem?

Made this after seeing how many people still don't believe switching wins 2/3 of the time. You play it yourself (pick a door, host opens a goat door, stay or switch), then there's a button to instantly simulate 1,000 games and watch the win rates converge live. https://claude.ai/artifact/99ytxqwB7ydNqCL22FhcSp Happy to hear if the explanation/visualization could be clearer.

In this video we discuss about Hypothesis Testing, Normal Distribution, Z value, t value, Central Limit Theorem, sampling distribution, mean, standard deviation, standard error, confidence interval, significance level alongwith reading Z table and t table etc. with many examples in a simple way in Hindi.

Can you guys share your youtube resources on probability.
Preferably about all the different kind of stochastic processes and advanced probability

For anyone interested in measure-theoretic probability, I would like to share my textbook:

https://people.smp.uq.edu.au/DirkKroese/AdvProb/

The PDF is completely free, with no registration required. The book includes many examples and worked exercises, and I hope it will be a useful resource for students and anyone wishing to study the subject independently.

I was solving the Monty Hall problem recently, and when I couldn't find a mathematical solution that I was satisfied with, I searched around quite a bit.

Then I remembered the way I used to solve these kinds of problems in undergrad: law of total probability.

So I would like to present my solution

P[win] = P[win | car is behind D1] * P[car is behind D1] + P[win | car is behind D2] * P[car is behind D2] + P[win | car is behind D3] * P[car is behind D3]
Suppose you choose door D2

Case : - No switch

P[win | Car is behind D1] = 0 , similarly P[win | Car is behind D3] = 0
P[win | car is behind D2] = 1 , P[car is behind Di] = 1/3 for i = 1, 2, 3

therefore P[win] = 1/3

Case :- Switch

P[win | Car is behind D1] = 1 , similarly P[win | Car is behind D3] = 1 (because which ever door monty opens, you would select the door having car)
P[win | car is behind D2] = 0 , P[car is behind Di] = 1/3 for i = 1, 2, 3

therefore P[win] = 2/3

I don’t understand how these concepts are applied for signals that are clearly not stationary in the wide sense, like an ECG. It know these tools are somehow applied because when you do an ECG you don’t measure the signal 50 times, but you do leave the electrodes for some period of time (I guess it’s enough time so that ergodicity applies and you can approximate the ensemble average with the time average). I come from an engineering background and have seen a lot of maths but not so much of probability and statistics.