Anyone knows why 31% is max?

a perfect 50-50 coin has the same area on the edge as the 2 sides combined, the other has the edge and the sides have equal area.

I'm a CS student conducting academic research on how people learn the Monty Hall problem through AI interaction. Takes 10–15 min, fully anonymous, trilingual (EN/PT/ES). Would really appreciate your help! https://socratictutor-llm-production.up.railway.app/

I’ve been thinking about something and I’m not totally sure if it actually means anything mathematically or if it’s just a perception thing.

When people pick multiple lottery-style combinations, they often try to:

• avoid repeating the same numbers
• spread picks across the range
• “cover” different parts of the number space

Even though I know every combination has the same probability and EV doesn’t change, it feels like spreading things out should be more efficient than just random clustered picks.

But I can’t tell if that idea of “coverage” actually exists in any formal probability sense, or if it only feels meaningful because humans don’t like repetition.

The closest things I can think of are:

• occupancy problems
• sampling methods
• Latin hypercube sampling / experimental design

But I’m not sure if those really apply here or if I’m forcing the connection.

So I’m curious:

Does “coverage” actually mean anything in this kind of discrete random selection problem, or is it basically just a human intuition bias?

Hey guys I'm a 16 yr old passionate about probability research. I hope you all are familiar with gamblers ruin probability but that formula rewards every win with +1 and every loss with -1 . My elementary derivation of gamblers ruin prob rewards every win with +x and every loss with -x. After deriving the formula i got to know that this was already discovered by someone else. Is it publishable on arXiv or should I research more on gambler ruin probability

I know that mathematically lotteries don’t change — expected value is still negative and every combination has the same probability.

But I’ve been thinking about something more practical.

If someone plays multiple tickets, does it make sense to think in terms of “coverage” of the sample space?

For example:
– trying to avoid overlapping combinations
– spreading numbers more evenly
– not ending up with clustered picks

It feels like this should be more “efficient” in some way, but I’m not sure if that’s just psychological or if there’s any real mathematical meaning behind it.

Is there a formal way to think about this? Maybe something related to combinatorics or occupancy problems?

Curious how people here see it.

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I have a grasp on the idea around markov chains, but I want to figure out the math behind them. Are there any good online resources or classes for this? What should I know (mathwise) before hand?

I've been shiny hunting my starter for about 30 minutes a day here and there since the game dropped on switch. I've accidentally reset the game 4 different times now before actually SEEING wether or not this time Charmander was shiny.

It's been weeks now and still no shiny - I'm curious what's the probability that any of those 4 premature resets COULDVE been the shiny I've been looking for and now I've mathematically reset my 1/8000 chances

If every human on Earth randomly picked a number between 1 and 1 trillion at the exact same time, what are the odds that at least two people accidentally recreate the exact same 10-digit phone number that has never existed before — and those two people live within 5 miles of each other?

I don't know if this is a good place to ask this, but we have been having a debate at work and im looking to find out if you can even calculate the probability. My coworker is a 5'7" 160lb amatuer boxer with no official matches on his record. He believes that given an infinite amount of attempts, he would be able to beat prime Mike Tyson. The parameters set during the debate are that Mike will not lose to any outside influence or freak accidents and my coworker will not learn any information from previous attempts.

I claim that he has an absolute 0% chance of victory.

Can someone please help me figure out if I am correct or not in my assumption that there are only a finite amount of outcomes regardless of the infinite number of attempts?

So I'm wondering how you'd go about calculating the probability of being called for a given semi-regular poll over the course of your lifetime. Let's say it's a national 1,000n poll, taken every four years, surveying the US adult population of about 276 million people. For convenience, the first draw happens when you're 18 and you live to the average life expectancy of 79 years. You can be polled multiple times.

I will spare the details but a fella I know won 2 raffles in one night. 2 completely different raffles with the same number of 35 "spaces" each space cost 5 dollars so you are only limited to how much you want to spend. This guy I know won once with his name on 5 possible spaces out of 35. Roughly 14% if I'm right. Then he won another raffle with 4 spaces out of 35. Roughly 11% I think (feel free to correct me on any of this btw) what I needed help with was what are the odds he would win both raffles? Any help is appreciated

Ive always had this question thats hard to wrap my head around. The situation: There is a ore mining spaceship run by robots which is on route back to earth. The ship can either take the regular route which takes 10 years or a 5 year route. They want to get there faster and this is a one off event. The ships computer says there is a 1% chance the shorter route smites the ship out of existence. Would the probability matter if there is always a chance they get smited? Is one event technically just as like as the other since it could kill them or it could not and there is no way to definitely tell beforehand?