IQ (moved subtopic)

fwiw after some prompting copilot agrees with DS

New Conditions:

Painting Requirement:

Each slat must be painted by either entering or exiting from both sides (left and right).

Starting Slate:

The starting slate (double zero) is already considered painted because it touches two slats.

Impact on Probability:

Increased Complexity:

The ball must now enter or exit each slat from both directions to be considered painted.
This means each slat needs to be visited in a way that covers both its internal sides.

Non-Uniform Visits:

The probability distribution of visits to each slat is no longer uniform.
Slats closer to the starting point might be visited more frequently initially, but over time, the random nature of the walk will still distribute visits more evenly.

Last Slat Painted:

The last slat to be painted (entered or exited from both directions) will depend on the specific path taken by the ball.
However, due to the random nature of the walk, each slat still has an equal chance of being the last one painted once the ball has had enough time to visit all slats from both directions.

Probability Calculation:
Despite the increased complexity, the fundamental principle of symmetry in random walks still applies. Over a large number of steps, the ball’s position will be uniformly distributed around the circle. Therefore, each slat has an equal probability of being the last one painted.
Thus, the probability that any particular slat is the last one painted remains:
381​
This is because, in the long run, the random walk ensures that each slat is equally likely to be the last one painted, even with the additional requirement of entering from both directions.
If you have any more questions or need further clarification, feel free to ask!

Hmm I guess that makes sense. Still not quite as intuitive or elegant as I was hoping. I might write a sim for this for the toy 6 sector version and have a look at some of the patterns it generates just to get a better feel for what's going on.

Wait, this is wrong though. A slat gets painted once the ball enters it. If we start in slat 1 and flip heads and go to slat 2, slat 2 is now painted. If we flip tails and go back to slat 1, slat 2 remains painted. But the other side of slat 2 has not had the ball enter or exit through it.

If entering from either direction qualifies as fully painted it's just a random walk. If both sides of each number need to be painted it's as per above (ie it becomes a random walk anyway soon before 37 out of 38 had been painted both sides)

Right, it's a random walk, the question is essentially saying "show that the last slat to be painted is uniformly distributed". I'm not at all convinced the answer you posted shows this. You are not painting the sides, you are painting the slat. Sklansky was just using the ball touching the sides as an aid to help in following along with his logic.

Also, the AI answer is conclusory. It pretty much just states the answer without much reasoning as to what's going on. Intuitively we'd obviously expect the most likely last slat to be the one diametrically opposite the starting position.

Imagine you are blindfolded and walking around a circular track, taking random steps left or right. Over a long period, you will have visited every part of the track multiple times. The fact that you started at a specific point becomes irrelevant because your steps are random and independent. Each part of the track has an equal chance of being the last place you visit before you stop.

(this is what copilot gives you when you ask to give it in layman terms)

No, it doesn't. If the track has 36 slats and I take 1 step, there is a zero percent chance I end up in slat 18. If I take 18 steps, there is also an almost zero percent chance I end up in slat 18.

What the problem is saying is that you take some steps and stop when you have visited each slat at least once. That is the key here, otherwise it wouldn't work. With any given finite number of steps, slat 18 will always be less likely than slat 1. The key differentiator here is that rather taking some given number of steps, we only stop when we have reached each slat.

A more helpful way to look at it is that to end up in slat 18, you have to throw 18 more heads than tails. In order to end up in e.g. slat 3, you have to throw 3 more heads than tails. The chances of this are higher than throwing 18 more heads than tails in any given number of throws, no matter how large. So, the probability of just visiting a given slat in N throws is not uniformly distributed. The difference between this and Sklansky's problem is that N is not a fixed number in his problem, N is the number of throws it actually takes once the experiment is conducted.

What you are saying is as the number of steps approaches infinity, the probability visiting every slat approaches 1 and the probability of stopping on a given slat approaches 1/number of slats. That is not the same thing as saying that the probability distribution is uniform for a finite number of steps.

If you are going to work with a smaller wheel since there is nothing special about 37 slats, then bring it all the way down to the starting slat plus three more. Pretend its a clock where the ball starts at 12 and the three slats are at 3,6, and 9.

Now lets figure out the chances that 3 is the last one to be painted. Call those chances x. If the coinflip puts the ball into three it has no chance. If the next two flips are 9, then six (1/4 chance) 3 is the nuts. If the next two flips are 9 then back to 12 (again a 1/4 chance) 3's chances revert back to x.

X = 1/4 times 100% plus 1/4 times x

3/4 times x = 1/4

3x = 1

x = 1/3

But if 3 oclock has a one third chance, so must 9 oclock. Which means 6 oclock has the remaining one third chance.

The math stuff is cool, but I'd be more interested in hearing what people have to say about writers and how they'd go about estimating their intelligence. I find the insights and abilities of literary savants like Shakespeare, Goethe, Kierkegaard and Huxley to be equally, if not more impressive than, let's say, Einstein. Something about a built-in and extensive understanding of people combined with a talent for words.

I had also used the clock analogy with 4 "hours" before going with the pie chart analogy, since it seems less natural to think of the clock in terms of sectors and more natural to think of it in terms of points, and I kept making fencepost errors.

But more importantly, I didn't see how the bolded follows from what preceded it. 3's chances were x before 9 was painted, so there were 3 possible candidates. You are saying that 3's chances are still x once 9 has been painted and there are now only 2 possible candidates. I think you need to substantiate this. I could not substantiate it with any intuitive reasoning which is why I went down the road of reasoning in the way that I laid out in my earlier post.

I think you might be conflating x with two separate probabilities here. The probability that 3 ends up being last from the initial state of the wheel, and the new probability that 3 ends up being last with the new state of the wheel (9 having been painted combined with the probability that 9 had been painted and we've gone back to 12), and then essentially undoing that conflation to arrive at the result. Would you agree that if we just start the experiment with the initial state of 9 having been painted, the probability of 3 is y, and if we start with the experiment with the clean initial state, the probability of 3 is x, and x != y?

Yes. Got careless again. 3's chances is 1/4 plus a1/4 squared plus 1/4 cubed etc .It doesn't revert. Its a bunch of 9.12s culminating in a 9,6. But the answer remains 1/3.

I'm going to code a sim up for this at some point when I'm bored at work and look at the patterns it generates, as I still don't have an intuitive feel for what's going on here.

What was that thing about bearded guy pictures all about? I have no idea what that was in reference to.

What is probably bothering you, perhaps subconsciously, is that normally if there are 37 attempts at a 36-1 shot there will often be zero successes or more than one. But if there cannot be more than one success than y tries at a 1/x shot will result in that one success happening y out of x times. And if y=x then it will always happen. In this case there will always be exactly one success. So if each slat will have its 36-1 shot guaranteed, then 37 slats results in exactly a 100% chance that one of them will be the last one painted.

Not at all. What is bothering me is that if you just did some fixed number of coin flips, the probability of visiting each slat decreases with its distance from the starting slat. This is intuitive because you need to flip N more heads than tails (or 37 - N more tails than heads) to reach slat N. Moreover, if you visit slat N, then you have already visited all the slats either 2-N or N-37. So quite clearly, you are more likely to visit (and therefore finish on) slat 5 than slat 18 in any given finite number of flips. If I told you that I was going to flip 100 times and pay you even money on the ending number, you would probably do fine betting 1, 2 and 37, and not so well betting 16,17,18 and 19.

But essentially, because we are not doing a fixed number of flips but rather we are flipping until we reach each slat at least once, we are saying that the probability distribution of the ending slat now becomes uniform. So even though we will put many more coats of paint on slats closer to the start than slats further away, our actual ending slat is uniformly distributed. I find this is very counterintuitive. If you read my initial attempt at the solution, I have tried to find the most intuitive explanation for this that I could.

so this is really interesting because i recently was watching the original shogun and in that they have a scene where they draw straws and the captain is unconscious so they decide he'll get the last draw

upon watching, that felt really unfair, to the point that unconscious man would be most likely to not get the bad straw because everyone prior to him has to avoid drawing the short one

so out of curiosity i ran a monte carlo sim on it and sure enough, while incredibly unintuitive, everyone indeed had an equal chance of drawing straws regardless of order (if there's no cheating) whereas even the man who picks 2nd to last has at worst a 50/50 chance but the captain always loses if it comes to him so it evens out

but that's operating on a finite set, if we didn't remove straws as they were drawn then going later would indeed be a massive advantage and this kind of feels more akin to that since we are constantly resetting

ie the slats closest to the starting position are 50/50 in the first instance, but then the other side which was faded only needs 2x in a row the other way to also get hit itself (or just 1 more of the other side) ie heads + tails + tails or the inverse will knock out both right away

but i also felt similarly about the captain drawing straws so perhaps i'll build a monte carlo sim for this as well

This is a giant segue, but IMO interesting.

Was Jefferson scum just for owning slaves, or for having enough self awareness to realize it was a bad thing, but owning slaves anyways?

If it is just the former, I would say the majority of humanity through time has had no moral compulsions against slavery. So if anything "slavery" seems to be the norm and modern day Western societies rejecting slavery, and forcing much of the rest of the world to stop slavery, the exception.

If you are arguing modern liberal Western culture is great and the rest of most of the cultures of the world, and through antiquity, are scummy than I wont disagree with you. But that seems harsh.

You had to run a monte carlo simulation to satisfy yourself on this point? It seems incredibly intuitive to me unless I am misunderstanding what you are saying.

ran into a hiccup, i can easily monte carlo to find out how many coin flips is needed to get to whichever number last but then taking those results and using them as inputs and putting them into a separate monte carlo to find out the frequency it indeed lands on 24 last etc is something that excel has trouble with (or at least the way i do it in excel has trouble)

gotta get back to work but will resume this later and try to figure out how to solve this challenge of inputs

anecdotally though looking at the first few dozen results, the final number is usually quite far from the initial starting point and only takes a few hundred coin flips to make it there

Yes the straw thing is totally intuitive. I struggle with the wheel. Would need to work it out on a small one and try to get a feel for it

Almost all people in the modern United States reject slavery on a variety of grounds (including obvious moral grounds). It is indisputably true that some of those people would have been willing to own slaves if they had grown up during Jefferson's time. The same is true for a lot of issues. For example, it would be abhorrent by today's lights to argue that women should not be allowed to vote. But it would have been quite trivial to find people who held that view in 1780, including a great many women.

In many cases (maybe not in 18th century USA, but historically), the alternative to slavery was automatic death (in USA it would have been "you aren't in the USA"), so i can see plenty of you guys actually being definitely pro-slavery in those (very common, standard) historical scenarios, same as most of you seem to prefer life in prison to the death penalty.

My city banned it in 1257 (they also abolished indentured servitude)

sounds like someone is angling to overtake me in the standings 😀

to clarify, i assumed it was probably fair but something was gnawing at me that perhaps the unconscious dude was given an unfair advantage so i decided to test it

what initially set off the entire thought exercise wasn't that someone was going last, because someone was always going last, the main impetus was the absurdity that one of them was basically half dead already and it made most sense to sacrifice that person for execution instead of someone who was not comatose and then on top of it there was a fleeting suspicion that perhaps he'd have a slight advantage as well

this is also why the answer for many questions is done not through figuring out a precise formula to determine the result but rather building a simulation and then running that thousands of times and tracking the results

also, at least in sportsbetting, there's a very real and known phenomenon where perceived edges are never matched in reality - what you believe is an edge of 6% will instead yield a 3% roi etc etc
- i have not exhaustively searched for answers on why or how this is the reality so i generally like to test perception wherever possible, especially since my livelihood depends upon it so i could envision a situation where 10 people drawing straws would not have 10% equal chance but rather a sliding scale of 10.3% to 9.7% which gave a small edge to those in the back despite that in theory it should all be equal

ie even doing this monte carlo to test this roulette thing, i'm sure in the near future i'll find an angle that can be specifically tackled using the methodology i learned/developed in order to tackle this exercise

that stuff is also not as well understood to the public as it is to you nor even to the sportsbooks who regularly misprice things like "series won in 7 games" because they didn't properly calculate it

And for one brief shining moment he did. And then he has to go and make a poor analogy equating enslaving with disenfranchisement while basically showing the error in the same post by noting that many women themselves thought women shouldn't vote, which would virtually never be the case with slavery. So you are probably safe for now. However, you are on very thin ice because your post describing how a simulation sometimes gives you better advice about reality than theory does not apply to pure math problems which the roulette question is.

You also are on thin ice because I have a soft spot in my heart for those who immediately start thinking about techniques to theoretically solve an interesting math/logic problem even if they struggle at first and even if they are the two nuttiest posters on this website and even if they think I am a grifter and even if they think I am evil for leaning towards utilitarianism. So you may be caught from much further behind than Roccoco.