Hut luck

[Unimatrix11]

So let me try to sum this up: In the 1000x 3-times-in-row-heads experiment, you´d also, before anything starts, place your bet on 163 (sorry, i indeed made a mistake there). Then if we were to take a break after toss number 200 and it had already occured 100 times, and you were given the chance of adjusting your bet, you´d go like this: 100 (already occured) + 800 (remaining tries) * 1/8 (probility of hit each time) = 200, right ?

EDIT: Oh, you just said that... i see...

EDIT2: I still find it hard to wrap my mind around this, and even though i think i get the logic behind this (it goes pretty much like: the experiment always starts now - it has no past), i will probably keep making the mistake, that i´d always expect a ground equally covered with snow, rather than the flakes all being piled up in one spot... But if people are bothered with my ´odd´ thinking, we can just leave it here. I always tried my best during this discussion, not to come off like i knew ´better´. If i still left the impression that i wanted to proof 1+1 was not 2, then i guess i have to apologize.

I guess i was on my way there anyways, because as i see it, when i pondered that maybe the mass-experiment cannot be split, it was about the ´it always starts now´. In fact, it seems like it can either be completely in the future, or completely in the past, but not split between the two, for it to have constant probabilities. Before you go hut-popping it´s totally okay, to say popping 8 techs in a row now has a 1:10^8 chance. But as soon as you start that changes. When you hit something else, it immediately drops to zero. When you hit the first tech, it´s 10^7 and so on. Only after you have actually done it, it is (or now: was) 10^8 again (yesyes, i know it is 1, because its past now, but you say "the chances were 10^8 for that to happen"). I still find it remarkable, that the chances of popping 8 techs then isnt always the same, namely after popping 7 techs its a full 1:10. I see now, that this is perfectly true, even trivial in a way, since it must have come closer to reality by the 7 previous tech-poppings. You´d say "i am so close now". Yet again, you must admit, that it is not trivial. that popping the 92th tech after having popped 91 before takes the same amount of luck as popping the first one - that it makes no sense to say "if i pop a tech this time again, it will be my fifth in a row and thus i would be really lucky." I, at least, would (and did) have an issue with a statement like "well, not luckier than the first time" as a response. The former statement treates the 4 previous tech poppings as if they hadnt occured yet, by taking them into account (see, how people can find that wierd?), while the later realizes that they already happened and thus does not take them into account, which takes more information from reality and thus is more correct.

And yet: If we had no way of knowing, how high the chances were for a tech to pop, we could still figure it out by a long series of tries. We would deduce the probaility from past events, while when we are actually conducting a series of tries, we may not deduce anything from past events for the next one. I think the key here is, that we´d have to do such an awful amount of tries (i´d say we had to continue until ´it´happned at least 100 times, then devide that by the number of tries), that by that alone we´d kill randomness. Randomness then is the abundance of possible outcomes in a limited amount of tries. When bell-curves start to become true, is when the amount of tries exceeds the amount of possible outcomes to a certain degree, so that everything that is possible will actually occur. When everything goes from possibility to reality, then nothing is random anymore.

Still i have to say that the number of all civ-games ever played comes ´close enough´ to infinity for me, to put some faith in a ´standard´outcome. Yes, rah´s lucky day didnt change anything for anyone, but i´d simply not expect it to happen a second time, while i would have expected it to happen once. This could equally well be explained, by saying that if the experiment started now (we have to), there is less games left for it to occur again. While for the total amount of games ever played, i´d expect it to happen once, i´d not expect it to happen for half that amount of total games (asuming one half of all civ games ever played have already been played) to happen at all. Now this has nothing to do with rah´s hitting it, but merely with the lower amount of tries left. A more accurate statement then would have been: "As i dont think anyone will have that luck for the remainder of the civ-games to be played (regardless of the ´rah-incident´), it is good to know that it at least happened once to anyone ever, for thats what i´d expect it to, given the chances and my estimate that civ will be played 100.000.000times in total, and about half have been played already."

Final word (for now ): The nature of ´Randomness´, ´Infinity´ and the difference between past and future are among the concepts that present the human mind with the toughest nuts to crack. I still believe, if anything about them seems trivial, then its probably because there are too many answers and to few questions in the mind of the thinker. For example, if we are talking ´infinite tries´, wouldnt that mean, that any possibility, no matter how unlikely to happen, would also occur infintetively often? Does ´randomness´ truely exist at all, or is it merely lack of information? If we knew everything about the present and the past, would we (not) also know everything about the future? Only if not, there would be true randomness, right? Randomness is such a hard thing to tackle actually, that gods are born from the human desire to eliminate it. At least i find no other way, to comfort my mum when she is mourning, repeatingly asking "Why?". "Bad luck" is such a dissatisfactory answer, just like any that draws its final reasoning from randomness, that humans will always try to push its existence (should it truely exist) out of reality, by finding some trick that makes things more understandable and predictable. I guess thats what i was trying in a way. So from that point of view, maybe you find it easier to forgive all my rambling (including this one).

[snoopy369]

Albeit your numbers are wrong in that example, yes. I'd bet on 125 before hand, but if it was 100 after try 200, i'd bet on 200 after 1000 (1/8*800 + 100). After I checked to make sure the coin wasn't a bad coin of course

[Yosho]

Unimatrix11: Yeah, it's really, really wierd the way statistics work, and it dosn't really jive with the way your brain intuits stuff.

Here's an even wierder example; it took me a long time to wrap my brain around this one:

You're on a game show. You have to pick one of 3 boxes; one of the 3 boxes has a million dollers in it, the other two are empty. You reach out, and pick a box.

Now, before you open it, the host suddenly gives you a choice. He shows you one of the other two boxes, and you see it's empty. He now gives you the choice; would you like to switch with the other closed box, or keep the one you have?

Which one should you take at that point, or does it matter?

[Krill]

Monte Carlo

[Unimatrix11]

Originally posted by Yosho
Unimatrix11: Yeah, it's really, really wierd the way statistics work, and it dosn't really jive with the way your brain intuits stuff.

Here's an even wierder example; it took me a long time to wrap my brain around this one:

You're on a game show. You have to pick one of 3 boxes; one of the 3 boxes has a million dollers in it, the other two are empty. You reach out, and pick a box.

Now, before you open it, the host suddenly gives you a choice. He shows you one of the other two boxes, and you see it's empty. He now gives you the choice; would you like to switch with the other closed box, or keep the one you have?

Which one should you take at that point, or does it matter?

Yosho, now what i really find wierd about that, is that with this one i dont have a real problem at all. To me it´s clear that this has no impact on my decision other than to be happy about one of the other boxes being empty, for it means, that the 1M$ is in one of the other two, from which i chose one. I would see nothing there, that would make me reconsider my decision. In fact i dont see a possible way of reasoning, that would indicate to me that i should. So: Am i missing something ? What would be the train of thoughts that could make me reconsider here (even if it was incorrect) ? (or are you just making fun of me ?)

[Yosho]

Originally posted by Unimatrix11


Yosho, now what i really find wierd about that, is that with this one i dont have a real problem at all. To me it´s clear that this has no impact on my decision other than to be happy about one of the other boxes being empty, for it means, that the 1M$ is in one of the other two, from which i chose one. I would see nothing there, that would make me reconsider my decision. In fact i dont see a possible way of reasoning, that would indicate to me that i should. So: Am i missing something ? What would be the train of thoughts that could make me reconsider here (even if it was incorrect) ? (or are you just making fun of me ?)

(nods) That's what I thought at first to, but it turns out, the correct answer is you should ALWAYS switch boxes at that point.

You picked a box. There is a 1/3 chance the money in in that box, and a 2/3'ds chance it is not.

Now, at this point, the host gets rid of whichever box is empty. This DOES NOT CHANGE the fact that the box you picked still has a 1/3rd chance of being right; no matter which box you picked, the host would still have thrown away the empty box, so that does not give you any extra information about the box you already picked. However, if your initial choice was wrong, (a 2/3 chance), then the money must be under the remaining box.

In other words, if you keep the box you already picked, you have a 1/3 chance of getting the money. If you switch boxes, you have a 2/3 chance of getting the money. So you should always switch boxes. It's not at all intuititive, but it is true.

[Unimatrix11]

Arrrrghhh ! While i am pondering this, i came across the idea, to just check in at a mental hospital before its too late - would make things a lot easier for all of us ....

As i see it, it goes like this:

Before: 1 out of 3
After: 1 out of 2

I am assuming that our Harry Weinfurth (host of ´the price is right´ in germany - at least he was - i dunno of its still on) knows where the money is. Since after you picked one box, two are still remaining, which means one of those has to be empty, no matter what box you chose. He is able to open an empty box thats not in your hand no matter what. Actually he will do that anyways - even before your pick, you know he will do that (if you know the show ). We can assign a probability of that to happen at 1, even before he does it. So this doesnt tell me anything about my box (as you say), but it doesnt tell me anything about the remaining box either. From intiution, that suggests, that it can have no impact on the probabilities of the second choice of yours.

Anyways. I think (!), with your reasoning, you are actually doing the same ´mistake´, that i did - you are splitting the experiment. But, as i understand it now (!), to be ´correct´, we always have to say ´the experiment starts now´ when we ask a question about future chances. In that sense the reasoning going along the lines ´if your initial choice was wrong (2/3 chance)...´ where the focus is on ´initial´ is not ´correct´. When we are asking, what chance there is for our box to contain the 1M$, we have to forget about all that happened before and only see the two remaining boxes. Both have an equal chance of containing the 1M$.

EDIT: As assessed with the tech-popping - tho this is a different kind of experiment, picking without returning the balls - if you are looking at the overall outcome probility from within the experiment (e.g. the probability of hitting 8 techs in row is different when already did 3 (1:10^5) from when you hit 7 (1:10^2) it changes, as you move along with it. So in fact, Harry throwing away one box, from this perspective (!) DOES change the probability of your initial choice being correct, namely from 1:3 to 1:2 (or for it being incorrect from 2:3 to 1:2 respectively).

It´s like with the guy who is showing you three cards on the table, one if it is an ace. He turns and swirls them around and afterwards asks you where the ace is (´Pronto Salvatore´ for the seasoned German TV-audience). Now instead of saying ´its this one´ when you point at card, you do it the other way around: You say: "well, i dont think its this one´ - he turns it and its not that one (2/3 chance). Now only two cards are left. Your next pick is 50/50. Looking overall to hit the right one this way is 2/3*1/2 = 1/3 - the same as if you went the ´normal´ way of ´positive´picking once. Point here is not so much the result, but the second multiplier (1/2) - if it had a different value, the way of picking would make a difference. But within three choices, saying "its not in A and not in B" is the same as saying "its in C" - so it cant make a difference. As the second multiplier is analogue to our second box-pick, i think i have just shown (if not even proven) that it has to be 50/50, in order the be mathematically ´correct´.

PS: Was this a test, to see if i got it ?

[Yosho]

Originally posted by Unimatrix11

I am assuming that our Harry Weinfurth (host of ´the price is right´ in germany - at least he was - i dunno of its still on) knows where the money is. Since after you picked one box, two are still remaining, which means one of those has to be empty, no matter what box you chose. He is able to open an empty box thats not in your hand no matter what. Actually he will do that anyways - even before your pick, you know he will do that (if you know the show ). We can assign a probability of that to happen at 1, even before he does it. So this doesnt tell me anything about my box (as you say), but it doesnt tell me anything about the remaining box either. From intiution, that suggests, that it can have no impact on the probabilities of the second choice of yours.

Anyways. I think (!), with your reasoning, you are actually doing the same ´mistake´, that i did - you are splitting the experiment. But, as i understand it now (!), to be ´correct´, we always have to say ´the experiment starts now´ when we ask a question about future chances. In that sense the reasoning going along the lines ´if your initial choice was wrong (2/3 chance)...´ where the focus is on ´initial´ is not ´correct´. When we are asking, what chance there is for our box to contain the 1M$, we have to forget about all that happened before and only see the two remaining boxes. Both have an equal chance of containing the 1M$.

Nope, that's simply not true.

You could do the experemnt yourself, if you wanted. Have 3 boxes. One box has something under it. Your friend knows which box it is. You pick one box. Your friend throws out the empty box. Now, 2/3'ds of the time, the object will be under the other box, and 1/3d of the time, the object is under the box you pick. Do it a bunch of times (say, 40 or 50 times), and you should see that for yourself.

The thing is, you can't actually break it up there, because you still don't know if the box you picked was the winner or not. Either the box you picked was the winner (still a 1/3 chance), or it was not; and if it was not, then the million dollers is under the remaining box.

Now, if the guy threw away one box and THEN shuffled the two boxes around so you had no idea which was which, and THEN you picked, yes, you would have a 1/2 chance of getting it right. However, that's not how it happened. In this case, you KNOW that the money is most likely not under the box you picked, so you KNOW it's probably under the remaining box.

It's not a new experement, it's still the same one, you're just given new information.

It´s like with the guy who is showing you three cards on the table, one if it is an ace. He turns and swirls them around and afterwards asks you where the ace is (´Pronto Salvatore´ for the seasoned German TV-audience). Now instead of saying ´its this one´ when you point at card, you do it the other way around: You say: "well, i dont think its this one´ - he turns it and its not that one (2/3 chance). Now only two cards are left. Your next pick is 50/50. Looking overall to hit the right one this way is 2/3*1/2 = 1/3 - the same as if you went the ´normal´ way of ´positive´picking once.

Right, that's true, but that's not the situation here.

PS: Was this a test, to see if i got it ?

No, it was just an mostly unrelated example of how un-intuitive statistics can be. The answer I gave is actually correct, and you even could test it experementally if you wanted, but it just dosn't feel right, does it?

[Unimatrix11]

I was in the middle of argueing your points, when i started to increase the box numbers - and then *bam* it dawned on me. Of course, Your are right, and it becomes pretty apparent, when you start thinking about 1,000 boxes instead of just three.

Thanks for teaching me and your patienced with it. Now i am going around, teaching others in the manner You did. Two people so far. Both thought 50/50 - just shows how unintiuitiv it is.

I would have answered yesterday, but i couldnt reach Apolyton - something wrong with the site yesterday ?