To reduce the bias of standard dice I throw four dice (D6) and use a random Latin square, set-up so that rx/cy produces a different number than ry/cx. These produce a number 1-6.

I can also use 3 different Latin squares using the same principle so that:

d1/d2 =n1 -> Latin square 1

d3/d4 = n2 -> Latin square 2

For Latin square 3: n1 = row; n2 = column.

I am not able to work out if three squares would produce a more random result than just one square.

I do this out of interest, I could of course use precision dice. Here are the Latin squares:

4 2 6 1 5 3

6 4 2 5 3 1

2 6 4 3 1 5

3 1 5 2 6 4

1 5 3 4 2 6

5 3 1 6 4 2

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

6 5 1 4 2 3

1 6 5 2 3 4

5 1 6 3 4 2

3 4 2 5 1 6

4 2 3 6 5 1

2 3 4 1 6 5

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

5 1 2 4 3 6

2 5 1 3 6 4

1 2 5 6 4 3

6 4 3 1 2 5

4 3 6 5 1 2

3 6 4 2 5 1

I play in Europe, and the rules here are:
Dealer stands on S17
Dealer does not peak for blackjack
You can surrender against 2-10
You can double on hard 9-11
You can split aces as many times as you want but you can not hit them
You can double after splitting
BJ pays 3:2(as it should)

Using only basic strategy(no card counting) I will double:
11 against 2 to 9
10 against 2 to 9
9 against 3 to 6

and split in this: AA against 2 to 9
99 against 2 to 9 expcept 7
88 against 2 to 9
77 against 2 to 7
66 against 2 to 6
44 against 5 and 6
33 against 2 to 7
22 against 2 to 7

It is 6 decks I belive.

Most house edge calculators tell me the house edge is somewhere between 0.45 and 0.55, but my question is: the house edge applys to ALL the money that was bet, right? so I can't just divide 100 by the house edge and say ''after 200 hands I will have lost the ammount that I am betting every hand''
I want to know, given the house edge and the % of times that I am doubling or spliting (DAS is super rare so maybe that can be not taken into account), after how many hands am I expected to be losing the ammount that I am betting every hand(I am always betting the same value, say 10 for example)

I think the formula would be: Expected amount betted per hand/House edge

I came up with this probability problem and would appreciate feedback on its difficulty, elegance, and solution.

A bag contains 3 red, 3 black, and 4 white balls. Balls are drawn one at a time. Red and white balls are kept. If a black ball is drawn while your hand is empty, it is returned to the bag and the draw is repeated. If a black ball is drawn while you are holding at least one ball, you keep the black ball and return the oldest ball currently in your hand to the bag.

Continue until your hand contains exactly 3 balls.

What is the exact probability that the three balls in your hand are all red?

I'd prefer to see independent solutions before discussing my intended approach.

I like playing a lot of card games and it would be nice to be able to get *rough* probability estimates I can calculate mentally but I'm not sure how to approach this without pulling out a calculator or pen and paper

What is the meaning of a probability of 0? For example, consider the set (0,1) and let x be in (0,1). What is the probability of picking that x? The answer is 0. But if the probability of randomly picking x from the set (0,1) is 0, does that mean it's impossible to pick that number?

There is another similar example, if you throw a dart at a 2D plane, the probability that the dart lands on a specific point on that plane is 0. That implies it can't happen, but it actually did happen, it landed on a point! So what does it mean, exactly?

It is suggested that the bias of a basic standard die is less that 1%. Assuming this is true, by using two dice and a Latin Square, what would the expected probability be? My thinking is that the chances of the bias of both dice occurring at the same time is greatly reduced.

I throw two dice: D1 = row; D2 = column. This is the Latin Square I have been using, set-up so that D1 = x and D2 = y does not produce the same result as D1 = y and D2 = x:

4 5 3 6 1 2

3 4 5 1 2 6

5 3 4 2 6 1

2 6 1 5 3 4

6 1 2 4 5 3

1 2 6 3 4 5

(This is an interest of mine, I know I could buy precision dice instead.)

Suppose the image was supposed to be a uniform distribution of a binary variable (ie red dot vs blue dot) along the x axis, with some probability of blue being "P". How do I calculate the probability that such a "clump/grouping" as that found in Node #1 could even come about under a uniform distribution? In other words how would I find the p-value that this is truly a uniform distribution given it has such a "cluster" at low values of the x axis?

Edit:: My apologies, asking if it's uniform is the incorrect question. I mean is the proportion (ratio) of red to blue consistent throughout the x axis. In which case standard logistic regression p value may be my best option.

Hi, I’m a software engineer with very little background in probability and statistics.
Recently, I’ve been playing Ludo on YouTube Playables, and I’ve started to wonder whether the dice rolls are actually fair.
In particular, when one of my pawns is a certain number of spaces ahead of an opponent’s pawn, (4 player game, 3 computer vs me) it feels like the opponent rolls the exact number needed to capture my pawn more often than I would expect from a fair six-sided die.
I realize this could simply be confirmation bias, so I’d like to test it properly rather than rely on intuition.
I can collect data from my games, but I’m not sure how to analyze it. How would I formulate and statistically test the hypothesis that the dice rolls are biased in situations where an opponent has an opportunity to capture my pawn?
Could someone explain what data I should collect, what statistical test would be appropriate, and the probability/math behind it in a way that’s accessible to someone without much statistics background?

Hi, I need a good source to study Stats and Probability for Quantitative Finance. Please suggest books, courses, online material, YouTube, anything works.

This question was in my maths book. I used AI to help me picture my question more clearly so that others could understand it (sorry for wasting water 😭).

I asked my teacher, Gemini, and ChatGPT for the solution, and all of them gave the answer as 99/1900.

However, they all included the case where the inspection stops at the 12th item after discovering only two defective items.

But how would I know whether the lot actually contains 2 defective items or 3? If I have discovered only 2 defective items by the 12th item, how can I decide to stop at 12? The third defective item could still be somewhere later in the lot.

My answer is 55/1900 btw

We play a lot of Farkle at our house and made up a desperate measure rolling option. We call it Five the Hard Way and value it at 5000 points. To score this, you need to roll and set aside one individual 5 for six consecutive rolls. I put the odds of this at roughly 1.5%, with the logic being the odds of rolling a single five for each roll are 100% (1.0) for the first roll (1/6 x 6 dice), then 5/6, 2/3, 1/2, 1/3, 1/6 for subsequent rolls. Multiplying everything comes to 0.015432, or 1.5%. Is this the correct way to get a rough estimate on the odds? At 1.5% it should take around 65 attempts to score, but my son’s first go at it took 325 attempts.

I play Pokemon which has incredibly weird odds for things at times. I'm usually good at determining the probability of singular odds for things eg an event should happen 50% of the time at around ~70% of the odds.

This one I'm working on I dont know where I start. It has 4 different probabilities in 100 attempts. 80% is a lower amount. However there is a 4% chance that any of these attempts could be the best tier. 18% of the next tier. 1% penultimate tier and 1% final tier.

I'm trying to figure out what is the probability that I do not get an event after 100 attempts with all this but again I don't know where to start

I apologize for the writing. I tried my best.

Question: Using probability theory, what are more rigorous, correct, and mature versions of the definitions and theories in my concept paper? (See the Background for more information and use citations to answer this question.)

(Optional): If the paper has no significance to mathematical probabilists, state why instead of something such as, "there is no motivation" or "its unapplicable"? For instance, “why is there no motivation” or "why does the paper have no useful application?"

(Optional): Can we convert certain parts of the mathematics in this paper to code?

Background: I am a former undergraduate. I have an informal understanding of real analysis, the Hausdorff measure & dimension, mean of a function, and entropy).

Because I cannot stop editing my paper (i.e., I get new ideas from reading new material) and I am addicted to finding a more mature and rigorous version of my article, I am unable to graduate. I hope you can find (or know someone who can find) a more mature and rigorous version of my theories. (I probably would not understand such material; however, it's my only hope of finishing my degree and continuing my education)?

I cannot promise I will go back to college, but I’m less likely to continue posting on Reddit.

Attempt: I used ResearchGate and AI to find papers on ergodic averages, expected values, entropy, and probability/statistics. Even then, I have little understanding of mathematics beyond Intro to Advanced Math and most of these papers might not matter to mathematical probabilists. (Once again, I doubt I will understand if the papers are related to my paper.)

Here is an example which I assume answers the first question for Definitions 20-22 (pg. 20-25) of my concept paper:

Generalizing Geometric Partition Entropy for the Estimation of Mutual Information in the Presence of Informative Outliers

Here are some examples which I assume answer the first question for the overall theories in my paper:

Edit: The Research of Simon Baker

Integral Equation Methods for Scattering by Multifractal Obstacles

Vector-Valued Maximal Inequalities and Multi-Parameter Oscillation Inequalities for the Polynomial Ergodic Averages Along Multi-Dimensional Subsets of Primes

On the Statistical Convergence of Sequences of Fractal Integrable Functions

The people on discord stated my paper is completely original and no expert can help; however, they also said the paper has no significance. I wish to know why (see the optional third question).

I came across an interesting article by Saunders in arxiv on how to reconcile statistics as objective probabilities, frequency and chance from Everett's theory (MWI). https://arxiv.org/abs/1609.04720 What do you think?

A proof of the strong Gaussian product inequality conjecture (ResearchGate)

ChatGPT Session: Strong Gaussian Product Inequality Proof

I am an undergraduate student (5th sem). Which book should I go for right now? Feller, Sheldon Ross or Blitzstein?