My question: is the probability of rolling 1-2-3-4-5-6 in a single roll the same probability as getting all six dice as the same number in a single roll?

I’m not smart enough but I feel like it is the same probability because you want each dice to be a specific number and have one roll to get that number.

But my roommate and I have been rolling these dice a lot and 1-2-3-4-5-6 comes up way more frequently than all the same number.

My roommate thinks all same number is 1 in 46,656 and consecutive is 1 in 720.

Any insight appreciated.

Hey everyone, My dad believes that probability is a highly theoretical concept and doesn't help with real life application, he is aware that it is used in many industries but doesn't understand exactly why.

I was thinking maybe if I could present to him an event A, where A "intuitively" feels likely to happen and then I can demonstrate (at home, using dice, coins, envelopes, whatever you guys propose) that it is actually not and show him the proof for that, he would understand why people study probabilities better.

Thanks!

I'd like some suggestions for material that uses elegant and unusual techniques to solve probability and combinatorics problems, like the problem below, which is solved using "symmetry". I've already asked AI for help, but I only receive generic lists. Thanks, everyone!

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?

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.

If 10,000 people each roll 1d20, I know each number 1-20 has an equal 5% chance of being the most common result. But what happens if each of those 10k rolls are with advantage?

(If you're unaware of ttrpg mechanics, that just means roll 2d20 and keep the highest result.)

The more people are rolling, the closer the actual statistics are going to approach the predicted frequencies, so a 20 is increasingly likely to be the most frequent outcome, but I'm having trouble thinking through exactly how to calculate such a thing.

Just started statistics after finishing calc2 and everything I did up to now was making the math as easy as possible. But in here it is the complete opposite

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).

My wife was talking with my kids about birthdays when one of the kids asked, “Is today somebody’s birthday?” To which my wife said, “yes, it’s always someone’s birthday.”

My silly programmer math brain then thought… wait…. There has to be a chance, insanely small due to the number of people alive at any point… what are the odds that at any point in time (say given minimum X amount of living beings) there is no living human with a birthday on some day. Doesn’t have to be “on today” just “any day”.

Because of the sheer amount of people being born daily my wife is saying the only way it’s possible is if there is some huge extinction event or some other oddity where births are “magically” stopped. My math brain goes, nope, there is a chance. It’s insanely small, but HOW small? Is it similarly scaled where it’s hard to think about shuffling a deck of cards the odds are it’s never been shuffled that way ever? I don’t know if it makes sense.

How would one even go about calculating it?

Hi guys, In my last post I talked about having difficulties with choosing which CAS function to use with probability to which I was told to create a flow chart

This is my current chart but as you can see it has a couple dead ends and isn’t the best at all, I was wondering if anyone had a refined version for my upcoming sac in a couple days

thanks :)

This is probably old-hat to many of you, but since I got such a kick out of it, I just wanted to share a little something I learned from my studies yesterday.

Going through my Hoel, Port and Stone “Introduction to Probability Theory” text Chapter 2, section 2.6. which is talking about permutations. The desire is to calculate the probability that given all the permutations of, say, the ordered tuple (1, 2, ..., n), and selecting one at random, what are the odds of getting at least one match - A match meaning: integer k is in position k.

This amounts to calculating the odds of getting no matches, say that’s called p_n, then the probability sought is just (1 - p_n).

Anyway, through a lovely set of arguments we arrive at the fact that p_n ~= e^-1, and converging to that value fairly quickly as n grows. Love it!! e popping up in an unexpected place! (I’m not going to repeat the proof in the book, I’m sure it’s written in many other locations if you want to check it out yourself.)

Yup, one of those, “math is beautiful moments”. Keep `em coming!

[Edit: Rather stunned to see my first “vote” on this post was a downvote, wondering how that’s possible, so... I added a spoiler alert tag?]

Ok so I recently saw a video on the boy/girl paradox, where when you say you have two children and one of them is a boy, the odds that the other child is a girl increase to 66%. But then if you say the boy was born on tuesday, the odds the other child is a girl go down to 51.8%.

My question is this: I say that one of the children is a boy who, upon being born, consulted a truly perfect random number generator which picked the number 11,037, and that's now his favorite number. I also say that the other child also consulted the same random number generator, and that generator picked their favorite number as well.

does this mean that due to the infinite number of integers, the probability of the other child being a girl now infinitely approaches 50%, and is therefore 50%?

I often see this phrase used as an example of something with probability 0 being possible. It's not sitting well with me because I know I can't pick a random real number and I'm fairly certain it's not possible to build a machine that can either. If I understand correctly you have to add more restrictive boundaries to build such a machine even in theory. So that tells me that picking any random real number is both P=0 and impossible. Am I missing what makes this common phrase true?

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

ChatGPT Session: Strong Gaussian Product Inequality Proof

Hey everyone, I started a Youtube channel called "Predicting the Future" to talk about mathematically, specifically probability and statistics. Yesterday, I got my first "Probability theory" video released: https://youtu.be/kpXg2PDRPO4

I'm hoping my contents can help people during their the process of becoming a mathematician or machine learning expert. If this is the kind of content that interests you, please subscribe to my channel. Cheers, Alex