When “drawing straws” is it better to be first or last?


Off-Topic Discussions

101 to 150 of 178 << first < prev | 1 | 2 | 3 | 4 | next > last >>

Speaking of probability stuff, in D&D 5e, they have an "Advantage" and "Disadvantage" system?
How much does this change the d20's roll outcome? It seems cumbersome.

.

Liberty's Edge

Electric Wizard wrote:
Krensky wrote:

* Throws nerf balls at jeff.

What did you say about engaging with the crazy up thread? Huh?

We are actually making a real joke about self-reference.

That you don't know about such a thing is not your fault, it's your teacher's fault.
They didn't know how to educate someone like you..

No, you weren't.

Jokes are funny.

Also, using the Liar's Paradox to try and show your intelligence is just sad. Didn't you study any modern philosophy, logic, or semotics in school?

Liberty's Edge

Grand Magus wrote:

Speaking of probability stuff, in D&D 5e, they have an "Advantage" and "Disadvantage" system?

How much does this change the d20's roll outcome? It seems cumbersome.

.

http://anydice.com/program/4e77


Krensky wrote:
Grand Magus wrote:

Speaking of probability stuff, in D&D 5e, they have an "Advantage" and "Disadvantage" system?

How much does this change the d20's roll outcome? It seems cumbersome.

.

http://anydice.com/program/4e77

Wow. It really has an effect. Maybe I should try it on for size?


True story time, kids!

Our 7th-grade teacher was trying to teach us probability. I was a well-known wise-ass in class. (Go figure!)

He asked 3 kids to volunteer to flip coins in front of the class and record their results on the board so we could see both trends (5 heads in a row, etc.) and overall probabilities (how many heads, how many tails). Of course I *had* to volunteer, and the first thing I asked was, "What do I write if the coin lands on its side?"

To which I got the response, "Shut up, NH."

So I dutifully went up, wrote down, "Heads" and "Tails" on the board, and flipped the coin.

And it bounced off the carpet, up against the wall, and ended up cleanly standing on its edge, leaning against the wall.

And the teacher gave me a detention, figuring I *must* have done it intentionally.

Yeah, right.


NobodysHome wrote:

True story time, kids!

Our 7th-grade teacher was trying to teach us probability. I was a well-known wise-ass in class. (Go figure!)

He asked 3 kids to volunteer to flip coins in front of the class and record their results on the board so we could see both trends (5 heads in a row, etc.) and overall probabilities (how many heads, how many tails). Of course I *had* to volunteer, and the first thing I asked was, "What do I write if the coin lands on its side?"

To which I got the response, "Shut up, NH."

So I dutifully went up, wrote down, "Heads" and "Tails" on the board, and flipped the coin.

And it bounced off the carpet, up against the wall, and ended up cleanly standing on its edge, leaning against the wall.

And the teacher gave me a detention, figuring I *must* have done it intentionally.

Yeah, right.

That is a great story! Did your teacher give you "Actual real world coins"? Or harvy dents?


Electric Wizard wrote:
NobodysHome wrote:

True story time, kids!

Our 7th-grade teacher was trying to teach us probability. I was a well-known wise-ass in class. (Go figure!)

He asked 3 kids to volunteer to flip coins in front of the class and record their results on the board so we could see both trends (5 heads in a row, etc.) and overall probabilities (how many heads, how many tails). Of course I *had* to volunteer, and the first thing I asked was, "What do I write if the coin lands on its side?"

To which I got the response, "Shut up, NH."

So I dutifully went up, wrote down, "Heads" and "Tails" on the board, and flipped the coin.

And it bounced off the carpet, up against the wall, and ended up cleanly standing on its edge, leaning against the wall.

And the teacher gave me a detention, figuring I *must* have done it intentionally.

Yeah, right.

That is a great story! Did your teacher give you "Actual real world coins"? Or harvy dents?

It's been a few decades, but my recollection is that he actually pulled together his massive salary and came up with three real live U.S. quarters for us to flip...


NobodysHome wrote:
Electric Wizard wrote:
NobodysHome wrote:

True story time, kids!

Our 7th-grade teacher was trying to teach us probability. I was a well-known wise-ass in class. (Go figure!)

He asked 3 kids to volunteer to flip coins in front of the class and record their results on the board so we could see both trends (5 heads in a row, etc.) and overall probabilities (how many heads, how many tails). Of course I *had* to volunteer, and the first thing I asked was, "What do I write if the coin lands on its side?"

To which I got the response, "Shut up, NH."

So I dutifully went up, wrote down, "Heads" and "Tails" on the board, and flipped the coin.

And it bounced off the carpet, up against the wall, and ended up cleanly standing on its edge, leaning against the wall.

And the teacher gave me a detention, figuring I *must* have done it intentionally.

Yeah, right.

That is a great story! Did your teacher give you "Actual real world coins"? Or harvy dents?
It's been a few decades, but my recollection is that he actually pulled together his massive salary and came up with three real live U.S. quarters for us to flip...

I'm just saying, be careful. It has be strongly asserted by some in this thread that "actual real world coins"

have Pr(Heads)=Pr(Tails)= 1/2, leaving no room for 'edges', or any bias.
So, if you Observed a coin landing on it's edge, then it could not have been an "actual real world coin."


1 person marked this as a favorite.
Electric Wizard wrote:

I'm just saying, be careful. It has be strongly asserted in this thread that "actual real world coins"

have Pr(Heads)=Pr(Tails)= 1/2, leaving no room for 'edges', or any bias.
So, if you Observed a coin landing on it's edge, then it could not have been an "actual real world coin."

Darn it! I need to find me some of those! Stanford researchers point out that such real-world coins as flipped by real-world humans don't actually exist!

(Interesting side note -- in the 1980's I heard that a casino in Las Vegas had tested quarters in a machine, flipping them over 100 million times, and found that heads were more likely by a slight margin (maybe 51-49), but my Google search-fu is only showing results of the Stanford paper showing that coins flipped by human beings are biased 51/49 in favor of the side that was up when the coin was flipped. Proving once again that experiment and theory are two different animals...)

EDIT: And for those involved in the actual debate, rather than "NobodysHome's story time", it is indeed quite valuable to consider probability from the point of view of an "ideal coin" that is exactly 50% heads and 50% tails. But, speaking as a math professor who has had to deal with his own share of smart-ass students over the years, it's best to say, "We're going to pretend this quarter is an ideal coin, even though we know it probably isn't quite right, and let's see what we come up with..."
Otherwise you end up in rather silly debates...

EDIT 2: And I'm not trying to dismiss the entire debate, but as a former math professor, I'll point out that theoretical probability is mind-blowing enough in and of itself (the birthday problem, the Monty Hall problem, etc.) without introducing the vagaries of the "real world" messing things up even further. I was just always careful to point out to my students, "We're making an assumption here that the coin is absolutely perfect. And we're also assuming that that's 'good enough' for what we're doing. If you get results that are totally different, the first thing to do is check that 'perfect coin' of yours..."


Electric Wizard wrote:
NobodysHome wrote:
Electric Wizard wrote:
NobodysHome wrote:

True story time, kids!

Our 7th-grade teacher was trying to teach us probability. I was a well-known wise-ass in class. (Go figure!)

He asked 3 kids to volunteer to flip coins in front of the class and record their results on the board so we could see both trends (5 heads in a row, etc.) and overall probabilities (how many heads, how many tails). Of course I *had* to volunteer, and the first thing I asked was, "What do I write if the coin lands on its side?"

To which I got the response, "Shut up, NH."

So I dutifully went up, wrote down, "Heads" and "Tails" on the board, and flipped the coin.

And it bounced off the carpet, up against the wall, and ended up cleanly standing on its edge, leaning against the wall.

And the teacher gave me a detention, figuring I *must* have done it intentionally.

Yeah, right.

That is a great story! Did your teacher give you "Actual real world coins"? Or harvy dents?
It's been a few decades, but my recollection is that he actually pulled together his massive salary and came up with three real live U.S. quarters for us to flip...

I'm just saying, be careful. It has be strongly asserted in this thread that "actual real world coins"

have Pr(Heads)=Pr(Tails)= 1/2, leaving no room for 'edges', or any bias.
So, if you Observed a coin landing on it's edge, then it could not have been an "actual real world coin."

No. You assumed that, without any justification when replying to my post.

I just didn't bother to correct you. It was a trap.:)

Mind you, I'd expect it to be pretty close to a 50/50 chance, with the chance of an edge landing being very low, but affected by the surface it's landing on. It might be possible to manufacture a coin with skewed odds, but I wouldn't expect it to be by much.

A skilled flipper could probably also influence the chances of heads or tails.


It's amazing how powerful Observed Data is. It drives the point home.
(What it really does is allow you to select the most-likely model.)


If only I had attended public school.


Another fun probability puzzle/paradox to work through is based on the reasonably well known idea that any string of numbers is equally likely if you roll a die and record the results in sequence, no matter how "special" one sequence appears. (So 135224653 has the same likelihood as 111111111).

The challenge is to try and reconcile that fact, with the intuitive (and this time correct) answer to the following question:

I've rolled a die twenty times and recorded the results in sequence. I've also made up a sequence of twenty numbers. Which did I roll and which did I make up:

12345612345612345612
22153416323641435333

They're each equally likely from a random process? Right?

Spoiler:
There are many implicit assumptions here, but cluttering probability questions by stating the assumptions is rarely helpful. Everyone knows what they are anyway and if you don't you can ask.


Oh man, this is awesome!


Steve Gedes wrote:
The challenge is to try and reconcile that fact, with the intuitive (and this time correct) answer to the following question:

Thats easy. Its the odds of sequential patterns vs not sequential patterns. "pattern-less gobbledygook" is more likely because its a MUCH larger catagory of answers than a recognizable pattern, even though the specific pattern itself is just as likely.

Liberty's Edge

You can't prove a sequence is random. You can only prove that it isn't random.


One of my absolute, all-time favorite problems is the "black hat paradox" that blows people's minds because it demonstrates that *any* information is significant.

On a shelf in a room are five hats, all identical except for the fact that three are black and two are white.

Three blindfolded men (all of whom are perfect logicians) are led into a room. Each has a hat placed on his head at random. The other two hats are removed. The blindfolds are removed so the men can see each other, but not their own hats.

The first man is asked, "What color is your hat?"
He responds, "I don't know."

The second man is asked, "What color is your hat?"
He responds, "I don't know."

The third man is asked, "What color is your hat?"
He answers correctly.

What color is his hat, and how did he know?

HINT: He doesn't even have to open his eyes to know the answer.

Solution:

His hat is black.

If the first man saw two white hats, he would know that only black hats were left, and therefore he would know that he was wearing a black hat.

CONCLUSION #1: Either the second man or the third man is wearing a black hat.

If the second man saw that the third man was wearing a white hat, then from conclusion #1 he would know he would have to be wearing a black hat. Since he doesn't know what color his hat is, the third man must be wearing a black hat.

CONCLUSION #2: The third man must be wearing a black hat.

The third man, having heard nothing except two, "I don't knows", can safely conclude that his hat must be black.

I love this problem because it demonstrates that *any* new information, even a simple, "I don't know," is enough to provide additional details of a problem and allow you to reach a conclusion.


BigNorseWolf wrote:
Steve Gedes wrote:
The challenge is to try and reconcile that fact, with the intuitive (and this time correct) answer to the following question:

Thats easy. Its the odds of sequential patterns vs not sequential patterns. "pattern-less gobbledygook" is more likely because its a MUCH larger catagory of answers than a recognizable pattern, even though the specific pattern itself is just as likely.

That's fine for a trend prediction but worthless in an individual blind trial.


Krensky wrote:
You can't prove a sequence is random. You can only prove that it isn't random.

You can't even do that. Random patterns can and do mimic order on occasion.

Given the premise that one is picked and one is the product of a d6 its easy to see which is far, far more likely to be the right answer, to the point that i'd risk my life on it for a pizza.


Steve Geddes wrote:
Another fun probability puzzle ...

You guys missed the actual question: "The challenge is to try and reconcile that fact, with the intuitive (and this time correct) answer..."

- - - - -

My first instinct is Chi-Square, but I'm not sure:

1: Chi("12345612345612345612") = 17.4
2: Chi("22153416323641435333") = 12.9

1 Loses, so 2 is your dice rolling. This seems very crude. I'll try a Runs Test next.

.

I have to go out for Dinner now, but I'll keep trying later this evening.
Steve, if you post an answer put it in a Spoiler Tag please.

Silver Crusade

1 person marked this as a favorite.
NobodysHome wrote:

One of my absolute, all-time favorite problems is the "black hat paradox" that blows people's minds because it demonstrates that *any* information is significant.

On a shelf in a room are five hats, all identical except for the fact that three are black and two are white.

Three blindfolded men (all of whom are perfect logicians) are led into a room. Each has a hat placed on his head at random. The other two hats are removed. The blindfolds are removed so the men can see each other, but not their own hats.

The first man is asked, "What color is your hat?"
He responds, "I don't know."

The second man is asked, "What color is your hat?"
He responds, "I don't know."

The third man is asked, "What color is your hat?"
He answers correctly.

What color is his hat, and how did he know?

HINT: He doesn't even have to open his eyes to know the answer.

** spoiler omitted **

I love this problem because it demonstrates that *any* new information, even a simple, "I don't know," is enough to provide additional details of a problem and allow you to reach a conclusion.

Three logicians walk into a bar. The bartender asks if they want a round of beers. The first logician says, "I don't know." The second logician says, "I don't know." The third logician says, "Yes!"

Edit: I happen to like "NobodysHome Story Time."


BigNorseWolf wrote:
Steve Gedes wrote:
The challenge is to try and reconcile that fact, with the intuitive (and this time correct) answer to the following question:
Thats easy. Its the odds of sequential patterns vs not sequential patterns. "pattern-less gobbledygook" is more likely because its a MUCH larger catagory of answers than a recognizable pattern, even though the specific pattern itself is just as likely.

It is relatively easy (with enough probability understanding) but it's more complicated than that. Your approach does lead to the correct explanation, but if we're going to use "patternless gobbledygook" as a criteria for determining the facts one step is going to be to define what we mean by "pattern-less gobbledygook". That's not as easy as it seems.


Steve Geddes wrote:
BigNorseWolf wrote:
Steve Gedes wrote:
The challenge is to try and reconcile that fact, with the intuitive (and this time correct) answer to the following question:
Thats easy. Its the odds of sequential patterns vs not sequential patterns. "pattern-less gobbledygook" is more likely because its a MUCH larger catagory of answers than a recognizable pattern, even though the specific pattern itself is just as likely.
It is relatively easy (with enough probability understanding) but it's more complicated than that. Your approach does lead to the correct explanation, but if we're going to use "patternless gobbledygook" as a criteria for determining the facts one step is going to be to define what we mean by "pattern-less gobbledygook". That's not as easy as it seems.

Especially since, unlike this example, sequences designed by people to appear random will often have less apparent pattern than truly random sequences.


Electric Wizard wrote:
Steve Geddes wrote:
Another fun probability puzzle ...

You guys missed the actual question: "The challenge is to try and reconcile that fact, with the intuitive (and this time correct) answer..."

- - - - -

My first instinct is Chi-Square, but I'm not sure:

1: Chi("12345612345612345612") = 17.4
2: Chi("22153416323641435333") = 12.9

1 Loses, so 2 is your dice rolling. This seems very crude. I'll try a Runs Test next.

.

I have to go out for Dinner now, but I'll keep trying later this evening.
Steve, if you post an answer put it in a Spoiler Tag please.

All those teachers you are worried about not teaching their kids have let you down again. Chi squared analysis is used to determine if an experimental factor significantly effects outcomes in a wild environment. So the fact that one series of numbers was constructed eliminates chi squared from being useful.

But on top of that you did it wrong. Chi value for series (1) is .0114 and for series (2) is .035, 5 degrees of freedom on a d6, chi threshold for 99% certainty is .554. As you can see the test values are far below the threshold of significance.

The other part that makes chi squared the completely wrong test is that order matters in this trial and chi squared is only useful for determining if the sample mean is significantly different from the expected mean.


thejeff wrote:
Steve Geddes wrote:
BigNorseWolf wrote:
Steve Gedes wrote:
The challenge is to try and reconcile that fact, with the intuitive (and this time correct) answer to the following question:
Thats easy. Its the odds of sequential patterns vs not sequential patterns. "pattern-less gobbledygook" is more likely because its a MUCH larger catagory of answers than a recognizable pattern, even though the specific pattern itself is just as likely.
It is relatively easy (with enough probability understanding) but it's more complicated than that. Your approach does lead to the correct explanation, but if we're going to use "patternless gobbledygook" as a criteria for determining the facts one step is going to be to define what we mean by "pattern-less gobbledygook". That's not as easy as it seems.
Especially since, unlike this example, sequences designed by people to appear random will often have less apparent pattern than truly random sequences.

What I like about this puzzle is that Ive seen the probability-educated arguing on the wrong side against the intuitively correct but maths-naive position. I used to post on a poker site (where people love showing how great at probability they are). There were pages and pages of responses to an OP which brought this up, laughing at him with an argument along the lines of:

"Any sequence is equally likely, you just assign some subjective "specialness" to the first series that isn't there. There's no reason to think either is more likely to be the random one."

It's not straightforward (in my opinion) to explain why we should have confidence that the first is manufactured and the second was the product of a random process. The initial thought (it's extraordinarily unlikely for the first sequence to occur randomly) is undercut by the identical observation about the second sequence.


That what it comes down to Steve, I could give you my opinion of which series is constructed and which was rolled; but I would have to know that one was constructed and one was rolled up front.

If it happened that the first series was presented as a random series amoug a group of random serieses(seresii?) then I wouldn't bat an eye.


Steve Geddes wrote:
thejeff wrote:
Steve Geddes wrote:
BigNorseWolf wrote:
Steve Gedes wrote:
The challenge is to try and reconcile that fact, with the intuitive (and this time correct) answer to the following question:
Thats easy. Its the odds of sequential patterns vs not sequential patterns. "pattern-less gobbledygook" is more likely because its a MUCH larger catagory of answers than a recognizable pattern, even though the specific pattern itself is just as likely.
It is relatively easy (with enough probability understanding) but it's more complicated than that. Your approach does lead to the correct explanation, but if we're going to use "patternless gobbledygook" as a criteria for determining the facts one step is going to be to define what we mean by "pattern-less gobbledygook". That's not as easy as it seems.
Especially since, unlike this example, sequences designed by people to appear random will often have less apparent pattern than truly random sequences.

What I like about this puzzle is that Ive seen the probability-educated arguing on the wrong side against the intuitively correct but maths-naive position. I used to post on a poker site (where people love showing how great at probability they are). There were pages and pages of responses to an OP which brought this up, laughing at him with an argument along the lines of:

"Any sequence is equally likely, you just assign some subjective "specialness" to the first series that isn't there. There's no reason to think either is more likely to be the random one."

It's not straightforward (in my opinion) to explain why we should have confidence that the first is manufactured and the second was the product of a random process. The initial thought (it's extraordinarily unlikely for the first sequence to occur randomly) is undercut by the identical observation about the second sequence.

Though it wouldn't be trivial to prove it, BNW's patternless gobbledygook is good heuristic as long as the pattern is obvious. As you get to longer sequences and subtler patterns the problem becomes harder.


BigDTBone wrote:

That what it comes down Steve, I could give you my opinion of which series is constructed and which was rolled; but I would have to know that information up front.

If it happened that the first series was presented as a random series amount a group of random serieses(seresii?) then I wouldn't bat an eye.

I would. In fact I'd assume he was lying if he said that. While any particular series is equally likely, a series we all recognize as a meaningful pattern is not anywhere near so likely as a series we don't.


thejeff wrote:
Though it wouldn't be trivial to prove it, BNW's patternless gobbledygook is good heuristic as long as the pattern is obvious. As you get to longer sequences and subtler patterns the problem becomes harder.

As you get to longer sequences, it becomes easier actually but yes.

I'm struggling mightily with Internet searching and I've forgotten the name (something like "Bolaffi numbers" - but that wasn't it). They are essentially a mathematically rigorous definition of the gobbledygook/ordered dichotomy and they are extremely rare once the number of digits reaches twenty plus.


thejeff wrote:
BigDTBone wrote:

That what it comes down Steve, I could give you my opinion of which series is constructed and which was rolled; but I would have to know that information up front.

If it happened that the first series was presented as a random series amount a group of random serieses(seresii?) then I wouldn't bat an eye.

I would. In fact I'd assume he was lying if he said that. While any particular series is equally likely, a series we all recognize as a meaningful pattern is not anywhere near so likely as a series we don't.

Law of Large Numbers disagrees.


BigDTBone wrote:
thejeff wrote:
BigDTBone wrote:

That what it comes down Steve, I could give you my opinion of which series is constructed and which was rolled; but I would have to know that information up front.

If it happened that the first series was presented as a random series amount a group of random serieses(seresii?) then I wouldn't bat an eye.

I would. In fact I'd assume he was lying if he said that. While any particular series is equally likely, a series we all recognize as a meaningful pattern is not anywhere near so likely as a series we don't.
Law of Large Numbers disagrees.

Not given a single sample.

Sure, if he'd rolled a sufficiently large number of sets and picked the one he wanted to present that's a different story. But that's essentially the same as just making the series up.

And if you present a sufficiently large set of random series to actually generate a recognizable sequence of that length, you'll have a vastly larger number of non-meaningful sequences.


thejeff wrote:
BigDTBone wrote:
thejeff wrote:
BigDTBone wrote:

That what it comes down Steve, I could give you my opinion of which series is constructed and which was rolled; but I would have to know that information up front.

If it happened that the first series was presented as a random series amount a group of random serieses(seresii?) then I wouldn't bat an eye.

I would. In fact I'd assume he was lying if he said that. While any particular series is equally likely, a series we all recognize as a meaningful pattern is not anywhere near so likely as a series we don't.
Law of Large Numbers disagrees.

Not given a single sample.

Sure, if he'd rolled a sufficiently large number of sets and picked the one he wanted to present that's a different story. But that's essentially the same as just making the series up.

And if you present a sufficiently large set of random series to actually generate a recognizable sequence of that length, you'll have a vastly larger number of non-meaningful sequences.

The law of large numbers accepts single cases. Because while you are presented with the one odd case in a series of random numbers, someone visiting LA will run into an old friend also visiting LA in a coffee shop at 2:42 pm and they will notice each other because they both ordered a tall skinny frap with no whip.


Steve Geddes wrote:


The challenge is to try and reconcile that fact, with the intuitive (and this time correct) answer to the following question:
I've rolled a die twenty times and recorded the results in sequence. I've also made up a sequence of twenty numbers. Which did I roll and which did I make up:

12345612345612345612
22153416323641435333

They're each equally likely from a random process? Right?

Ah ha! I got it.

The most direct way to reconcile the difference between these two sequences
is with First-Differences. We know, "The First-Difference of a random
sequence should again "look" like a random sequence."

First-Difference of 12345612345612345612 := 1111151111151111151
First-Difference of 22153416323641435333 := 0142135311325312000

The second one still "looks" random.

(I was planning to try a Runs Test and auto-correlation on the original
sequences, and see those tests would be even more striking on the
First-Differences. But I'm not going to anymore. I like this answer.)

.


Have you also played around with the Envelope Paradox? You might enjoy that too, if you havent seen it before:

I give you two envelopes and tell you one contains twice the amount of money as the other. You're invited to select one and you can keep whatever is inside. You choose, but before you open it I offer you the chance to switch:

"After all," I point out, "supposing your current envelope has $x in it, there's a fifty percent chance that you will gain $x. Contrast that with a fifty percent chance you will only lose $x/2. Clearly the expected value of switching is (0.5x - 0.25x) = 0.25x so switching is a no brainer".

Once you've switched, of course, I make exactly the same argument that you should switch back.


Envelope A 200

Envelope B 100

There is a 50% chance you will gain 100 dollars by switching
There is a 50% chance you will lose 100 dollars by switching

Not seeing a paradox here.


You don't know the amount. You have $x so the options are plus x or minus half x.

PS it's not really a paradox, of course, it's just presented as one. Many people struggle to explain it to themselves.


BigDTBone wrote:
thejeff wrote:
BigDTBone wrote:
thejeff wrote:
BigDTBone wrote:

That what it comes down Steve, I could give you my opinion of which series is constructed and which was rolled; but I would have to know that information up front.

If it happened that the first series was presented as a random series amount a group of random serieses(seresii?) then I wouldn't bat an eye.

I would. In fact I'd assume he was lying if he said that. While any particular series is equally likely, a series we all recognize as a meaningful pattern is not anywhere near so likely as a series we don't.
Law of Large Numbers disagrees.

Not given a single sample.

Sure, if he'd rolled a sufficiently large number of sets and picked the one he wanted to present that's a different story. But that's essentially the same as just making the series up.

And if you present a sufficiently large set of random series to actually generate a recognizable sequence of that length, you'll have a vastly larger number of non-meaningful sequences.

The law of large numbers accepts single cases. Because while you are presented with the one odd case in a series of random numbers, someone visiting LA will run into an old friend also visiting LA in a coffee shop at 2:42 pm and they will notice each other because they both ordered a tall skinny frap with no whip.

Different case.

If millions of people were posting strings of numbers asking "Is this random?", then yes, some of them wouldn't look random. But they're not.


thejeff wrote:
BigDTBone wrote:
thejeff wrote:
BigDTBone wrote:
thejeff wrote:
BigDTBone wrote:

That what it comes down Steve, I could give you my opinion of which series is constructed and which was rolled; but I would have to know that information up front.

If it happened that the first series was presented as a random series amount a group of random serieses(seresii?) then I wouldn't bat an eye.

I would. In fact I'd assume he was lying if he said that. While any particular series is equally likely, a series we all recognize as a meaningful pattern is not anywhere near so likely as a series we don't.
Law of Large Numbers disagrees.

Not given a single sample.

Sure, if he'd rolled a sufficiently large number of sets and picked the one he wanted to present that's a different story. But that's essentially the same as just making the series up.

And if you present a sufficiently large set of random series to actually generate a recognizable sequence of that length, you'll have a vastly larger number of non-meaningful sequences.

The law of large numbers accepts single cases. Because while you are presented with the one odd case in a series of random numbers, someone visiting LA will run into an old friend also visiting LA in a coffee shop at 2:42 pm and they will notice each other because they both ordered a tall skinny frap with no whip.

Different case.

If millions of people were posting strings of numbers asking "Is this random?", then yes, some of them wouldn't look random. But they're not.

You are confusing the law of large numbers with the law of truly large numbers.


BigDTBone wrote:
thejeff wrote:
BigDTBone wrote:
thejeff wrote:
BigDTBone wrote:
thejeff wrote:
BigDTBone wrote:

That what it comes down Steve, I could give you my opinion of which series is constructed and which was rolled; but I would have to know that information up front.

If it happened that the first series was presented as a random series amount a group of random serieses(seresii?) then I wouldn't bat an eye.

I would. In fact I'd assume he was lying if he said that. While any particular series is equally likely, a series we all recognize as a meaningful pattern is not anywhere near so likely as a series we don't.
Law of Large Numbers disagrees.

Not given a single sample.

Sure, if he'd rolled a sufficiently large number of sets and picked the one he wanted to present that's a different story. But that's essentially the same as just making the series up.

And if you present a sufficiently large set of random series to actually generate a recognizable sequence of that length, you'll have a vastly larger number of non-meaningful sequences.

The law of large numbers accepts single cases. Because while you are presented with the one odd case in a series of random numbers, someone visiting LA will run into an old friend also visiting LA in a coffee shop at 2:42 pm and they will notice each other because they both ordered a tall skinny frap with no whip.

Different case.

If millions of people were posting strings of numbers asking "Is this random?", then yes, some of them wouldn't look random. But they're not.

You are confusing the law of large numbers with the law of truly large numbers.

Ok. Now I have no idea what you're talking about.


thejeff wrote:
BigDTBone wrote:
thejeff wrote:
BigDTBone wrote:
thejeff wrote:
BigDTBone wrote:
thejeff wrote:
BigDTBone wrote:

That what it comes down Steve, I could give you my opinion of which series is constructed and which was rolled; but I would have to know that information up front.

If it happened that the first series was presented as a random series amount a group of random serieses(seresii?) then I wouldn't bat an eye.

I would. In fact I'd assume he was lying if he said that. While any particular series is equally likely, a series we all recognize as a meaningful pattern is not anywhere near so likely as a series we don't.
Law of Large Numbers disagrees.

Not given a single sample.

Sure, if he'd rolled a sufficiently large number of sets and picked the one he wanted to present that's a different story. But that's essentially the same as just making the series up.

And if you present a sufficiently large set of random series to actually generate a recognizable sequence of that length, you'll have a vastly larger number of non-meaningful sequences.

The law of large numbers accepts single cases. Because while you are presented with the one odd case in a series of random numbers, someone visiting LA will run into an old friend also visiting LA in a coffee shop at 2:42 pm and they will notice each other because they both ordered a tall skinny frap with no whip.

Different case.

If millions of people were posting strings of numbers asking "Is this random?", then yes, some of them wouldn't look random. But they're not.

You are confusing the law of large numbers with the law of truly large numbers.
Ok. Now I have no idea what you're talking about.

You're being Trolled dude.

.


law of truly large numbers
law of large numbers


QED

Liberty's Edge

BigNorseWolf wrote:
Krensky wrote:
You can't prove a sequence is random. You can only prove that it isn't random.

You can't even do that. Random patterns can and do mimic order on occasion.

Given the premise that one is picked and one is the product of a d6 its easy to see which is far, far more likely to be the right answer, to the point that i'd risk my life on it for a pizza.

What I get for discussing probability and entropy after having been awake for almost 50 hours...

I was conflating random number sequences and random number generators there. You can prove a RNG is not truly random, but you can't definitively prove that it is.


Steve Geddes wrote:

You don't know the amount. You have $x so the options are plus x or minus half x.

PS it's not really a paradox, of course, it's just presented as one. Many people struggle to explain it to themselves.

It looks like you're not setting your math up right if the numbers work out but your letters don't. What I'm "struggling" to explain is your math error, not the idea.

You effectively have X as a loss twice. It looks like between going from the word problem to the algebra you doubled the loss of X, framing what is either a gain of X or a gain of 2x as either a gain of x or a loss of X

Switch has a 50 % chance loss of X, and a 50% chance to gain X.


I've always liked this one:

Coupons in cereal boxes are numbered 1 to 5 and a complete set, one of each, is required to get a prize.
With one coupon per box, how many boxes on the average are required to make a complete set?


Electric Wizard wrote:

I've always liked this one:

Coupons in cereal boxes are numbered 1 to 5 and a complete set, one of each, is required to get a prize.
With one coupon per box, how many boxes on the average are required to make a complete set?

Assuming a random and equal distribution of coupons, 12.


BigNorseWolf wrote:
Steve Geddes wrote:

You don't know the amount. You have $x so the options are plus x or minus half x.

PS it's not really a paradox, of course, it's just presented as one. Many people struggle to explain it to themselves.

It looks like you're not setting your math up right if the numbers work out but your letters don't. What I'm "struggling" to explain is your math error, not the idea.

You effectively have X as a loss twice. It looks like between going from the word problem to the algebra you doubled the loss of X, framing what is either a gain of X or a gain of 2x as either a gain of x or a loss of X

Switch has a 50 % chance loss of X, and a 50% chance to gain X.

Its not whether you use numbers or letters that create the math error. (For example, if the chosen envelope has $100 there's a fifty percent chance I'll gain $100 if I switch and a 50 percent chance I'll lose $50).

Your final sentence is correct (if X is defined as the difference between the envelopes). But so is the sentence "Switch has a 50 % chance loss of Y/2 and a 50% chance to gain Y." (Where Y is the amount in the envelope you've chosen).

Like all pseudo paradoxes, the issue is resolved by analysing why the second formulation is incorrect, not by repeating the correct answer.


Steve Geddes wrote:
BigNorseWolf wrote:
Steve Geddes wrote:

You don't know the amount. You have $x so the options are plus x or minus half x.

PS it's not really a paradox, of course, it's just presented as one. Many people struggle to explain it to themselves.

It looks like you're not setting your math up right if the numbers work out but your letters don't. What I'm "struggling" to explain is your math error, not the idea.

You effectively have X as a loss twice. It looks like between going from the word problem to the algebra you doubled the loss of X, framing what is either a gain of X or a gain of 2x as either a gain of x or a loss of X

Switch has a 50 % chance loss of X, and a 50% chance to gain X.

Its not whether you use numbers or letters that create the math error. (For example, if the chosen envelope has $100 there's a fifty percent chance I'll gain $100 if I switch and a 50 percent chance I'll lose $50).

Your final sentence is correct (if X is defined as the difference between the envelopes). But so is the sentence "Switch has a 50 % chance loss of Y/2 and a 50% chance to gain Y." (Where Y is the amount in the envelope you've chosen).

Like all pseudo paradoxes, the issue is resolved by analysing why the second formulation is incorrect, not by repeating the correct answer.

BNW is looking at it from an omnipotent observer standpoint. Which is fine is you are designing the game, but it doesn't work as he describes because he has given himself access to hidden information.

The point is the player doesn't know how much is in each envelope so they don't know if they will get .5x or 2x by switching.

The odds of you switching to the big prize is exactly equal to your chance to choose the big prize in the first place. 50/50


Yes, to be clear - it's not a true paradox as it can be resolved (unlike "This sentence is false" or similar). Nonetheless, it did take 'them' quite a lot of effort to nut out exactly what the problem was. It was a reasonably productive discovery.

The challenge is to explain why the Expected Value calculation, as presented fails - not to establish what the real answer is (which is obvious enough).

Dark Archive

Pathfinder Adventure Path Subscriber

I think I see where you're going, BigD. It makes more sense to me if it's not couched as envelopes, but calling Heads or Tails. If you win the coin toss, you get $x. If you lose, you get $(x/2). I flip the coin, and conceal the result. Without knowing more information about the results of the contest, there's no statistical benefit to changing your initial decision.

101 to 150 of 178 << first < prev | 1 | 2 | 3 | 4 | next > last >>
Community / Forums / Gamer Life / Off-Topic Discussions / When “drawing straws” is it better to be first or last? All Messageboards

Want to post a reply? Sign in.