And when he (almost certainly) turns out to be correct, it will be a really cool observation!
Don't think of Hawking radiation as light-rays, but as "evaporation". i.e. black holes evaporate, and the process by which this happens is through Hawking Radiation.
Incidentally, this evaporation causes the Entropy of the universe to decrease (which is a good thing).
I have to vote for Shadowrun as well. I'm running a 4E Shadowrun game right now. The only thing that I find difficult is sometimes juggling the complexity of the Matrix, Magic, and Physical worlds at the same time.
I'm staring a group of n00bs in SR5. Can you point me to any
"Introduction to the Matrix and it's Mechanics" runs?
Psionics (noun)
1. what AI super computers use in place of magic, because computers are not alive and only living things can *do* magic.
2. Spock from >Star Trek< has psionics.
I roll my eyes and grab a yellow #2. It's fun to invent new games when
they work, but schlumping though ten bad ones for every good one is
sometimes trying. My feeling is this is going to end up a schlump.
"Ok. Hit me. What are we going to do here?", I ask big S.
Yet strange how it works when the geography is reversed - namely how close the Golden Hoard came to conquering most of Europe - damn that resin in the bows!
It's pretty interesting (or terrifying) only internal politics stopped the Hoard. But, *we* (humans) would still be the same today. In the modified-today, there would still be cars and video games.
DESCRIPTION: Calculate the probability of the "observed data" for a bunch of coins being flipped. (Note: 'p' ranges from 0 to 1 by 0.10, and then I normalized the output values just for fun.)
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?
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.
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.)
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."
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?
YES!!! So, you claim you understand the coin may be biased (I can see your AHEM,
as if you've been right all along. But you haven't.)
I have been, and still am. Your insane ramblings, insults, and unsupported pretenses at a superior intellect won't change that.
Quote:
This is called ignorance. Recognizing you have it is the first step.
Then, I tried to introduce you to the most trivial mechanics of decision making.
And you do it badly, because you think very poorly and very illogically. You think that any answer can go strait into the math when it can't: garbage in garbage out.
The odds of 6 heads in a row is only 1 in 64. You will see 10 less likely events every day of your life. You have probably seen enough coin tosses of 6 to have seen a group of 6 heads. If the coin/the tosser isn't fair then there is no mathematical way to predict the outcome- the actual odds would be based on the mechanism for cheating. A two headed coin would have a 100% chance of returning H, someone deliberately flipping the coin would depend on their skill level at doing so.
I do not know how to calculate the odds not because I'm a mouth breathing troglodyte but because the odds are completely impossible to know once you leave the assumption of an unbiased coin. Without knowing the mechanism for your bias your calculations are worthless.
No, just accepting standardized rules for statistical tests. So, again, if we ""assume ANY random coin then the likelihood of a weighted coin is extremely remote"" . So remote that the bias introduced will be 2 or 3 orders of magnitude below significant figures in the system. If the weighted coin distribution itself is weighted then that needs to be included in the premise of the statistical test, which it was not.
Quote:
You're talking about something intimately related, but different.
This is called choosing a Prior. You've chosen one that appears Gaussian ("likelihood of a weighted coin is extremely remote...")
and that suits yourself, and makes your post most-correct. (<-- you're trying to force a particular world.)
A Prior *must* be picked, so this is fine. But realize what you've done.
Indeed, circumstantial rules must be established. So I choose those which most closely mimick the world around me. It also mirrors how the question was framed.
So here's the question: Assuming a coin drawn at random from your change jar and observing a series of (H,H,H,T,H,H,H,H), what are the odds of the next flip being H?
Actual real world coins, not something created by the experimenter to trick you and not a two headed (or tailed) coin, because you've observed both results.
"Actual real world coins" --> Interpretation, "I have a fair coin, p=1/2." ( You've forced a value for Pr(Heads). )
This is a very easy question to answer.
IT'S 1/2 BECAUSE YOU FORCED THAT VALUE ON US.
No, just accepting standardized rules for statistical tests. So, again, if we ""assume ANY random coin then the likelihood of a weighted coin is extremely remote"" . So remote that the bias introduced will be 2 or 3 orders of magnitude below significant figures in the system. If the weighted coin distribution itself is weighted then that needs to be included in the premise of the statistical test, which it was not.
Quote:
You're talking about something intimately related, but different.
This is called choosing a Prior. You've chosen one that appears Gaussian ("likelihood of a weighted coin is extremely remote...")
and that suits yourself, and makes your post most-correct. (<-- you're trying to force a particular world.)
A Prior *must* be picked, so this is fine. But realize what you've done.
Indeed, circumstantial rules must be established. So I choose those which most closely mimick the world around me. It also mirrors how the question was framed.
YES!!! So, you claim you understand the coin may be biased (I can see your AHEM,
as if you've been right all along. But you haven't.), and yet you *still* posit
there is only one correct probability for a coin flip (one world).
Question: Why did you not account for this "harvy dent" coin in your answer above?
Ans: Because you don't know how to do it -- I know you think you have done so.
Putting this in your post --> (Baring someone pulling a harvy dent)
is saying, I know there is more stuff, but I don't know what to do with it.
This is called ignorance. Recognizing you have it is the first step.
Then, I tried to introduce you to the most trivial mechanics of decision making.
Don't kill the messenger. Blame your High School teachers, and maybe
your parents, They didn't teach you.
.
No, just accepting standardized rules for statistical tests. So, again, if we ""assume ANY random coin then the likelihood of a weighted coin is extremely remote"". So remote that the bias introduced will be 2 or 3 orders of magnitude below significant figures in the system. If the weighted coin distribution itself is weighted then that needs to be included in the premise of the statistical test, which it was not.
You're talking about something intimately related, but different.
This is called choosing a Prior. You've chosen one that appears Gaussian ("likelihood of a weighted coin is extremely remote...")
and that suits yourself, and makes your post most-correct. (<-- you're trying to force a particular world.)
A Prior *must* be picked, so this is fine. But realize what you've done.
YES!!! So, you claim you understand the coin may be biased (I can see your AHEM,
as if you've been right all along. But you haven't.), and yet you *still* posit
there is only one correct probability for a coin flip (one world).
Question: Why did you not account for this "harvy dent" coin in your answer above?
Ans: Because you don't know how to do it -- I know you think you have done so.
Putting this in your post --> (Baring someone pulling a harvy dent)
is saying, I know there is more stuff, but I don't know what to do with it.
This is called ignorance. Recognizing you have it is the first step.
Then, I tried to introduce you to the most trivial mechanics of decision making.
Don't kill the messenger. Blame your High School teachers, and maybe
your parents, They didn't teach you.
Also that the odds for 5 heads in a row and then a tails are the same as 6 heads in a row.
Interestingly, there are an infinite number of situations where this is Not true, and there is one case where it is true.
Personally, I would bet on Heads again. (But I'm a Bayesian.)
Its true.
The odds of hhhhhh is 1/2 * 1/2 * 1/2 * 1/2 * 1/2 *1/2
The odds of hhhhht is 1/2 * 1/2 * 1/2 * 1/2 * 1/2 *1/2
The odds of any particular combo you call is 1/2 * 1/2 * 1/2 * 1/2 * 1/2 *1/2. All heads just stands out more to us because its the only way to get six heads, whereas 3 heads and three tails can be obtained any number of ways like h h h t t t , h t h t h t , h h t t t h
If you're going to say i'm wrong, you need to provide an explanation.
It's not that what you have typed is wrong, it's that you've only considered one case
out of an infinite number of cases. You've picked the one case in which you are correct.
(I just realized some people call this wishful thinking.) To further illuminate
these murky waters (which absolutely hide tentacles), let me ask yet another question.
What is the chance one thing out of an infinite number of things is actually The Thing
happening now? (Ha! typing that amused me...) Seems likely the chance is low.
Anyways, to carry on with the story, let's put it another way, all you (we) know is five
heads have been observed. The "Big Question" is what is the chance of that happening
GIVEN *any* coin. (Dude, this is the Big Question, don't forget.)
To help answer, we need *another case* -- let us compare your world against another of
the infinite possibilities (in fact, there is no other way to do it.) In this other
world, there exists a gimmick coin, and it has heads on both sides! Dun dun daaah!!
Now, (*lightbulb* goes off) given that you have observed five heads in a row, which
world do you think you are in? Yours or tentacle-land.
That is, given an observed sequence of five heads in-a-row, which coin do you now think
is more likely to be in play? Is it more likely a fair-coin, with Pr(Heads)=Pr(Tails)=0.50,
has just flipped five heads in a row, or is it more likely a gimmick coin always
showing heads is being used. (Yep, now the ice is beginning to crack loose. Run! Tentacles...)
[Oh btw, as an aside, Google does this *all* the time. Google doesn't use If-Then
statements about webpages, all their decisions are Bayesian updates of a likelihood
function (or something like that, I forget the exact jargon now.
Maybe it's posterior something-something.)]
-= Lowering The BOOM!! or, "Welcome to the real world Neo." =-
--------------------------------------------------------
The Set-Up: I have two coins. One is fair with Pr(Heads)=0.50. The other is biased and
has Pr(Heads)=0.90. (Yay, I know above I used Pr(Heads)=1.0, but that's too boring.
Plus, I'm going to use these two coins again below for the codez.)
The New Question: You don't know which of these two coins I'm using (fair or biased),
but you watch me flip it five times and it comes up Heads each time. Would you bet me
$1 the next flip is Heads?
Now, Back to the Original Question at the top of the Page: If I flip a coin and it
comes up Heads 5 times in a row, should I next bet on Tails? It's due. . Remember,
you have no idea where I got my coin from.
Answer = absolutely do not bet on Tails.
*****
Now !!please realize!!, if the Question had instead been: "Given a fair coin with Pr(Heads)=0.50,
and If I flip this coin and it comes up Heads 5 times in a row, should I
next bet on Tails? It's due." Then, your analysis (above) is spot on correct. (*the
crowd cheers*) The reason, Because the set-up of this question places us into exactly
one of all possible worlds -- the world in which you are correct -- actually, the world
where Pr(Heads)=0.50. And, as they say, that is a story unto itself. (Ok, nobody says that, but the reason stands.)
*****
-= Come with me if you want to be Bayesian =-
---------------------------------------
"The parameter of a coin's flip-probability is a random variable."
Haha! I'm not really going to teach you Bayesian Statistics, because I don't think you
can do it. But, I'll leave you with a thought experiment.
Imagine we have 11 coins manufactured so their probabilities differ according to the
following. The coins are named 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and their probabilities
of flipping a Head is:
Pr_0(Heads) = 0.00 // this coin has tails on both sides
Pr_1(Heads) = 0.10 // read this as, the chance coin 1 flips a head is 10%
Pr_2(Heads) = 0.20 // read this as, the chance coin 2 flips a head is 20%
Pr_3(Heads) = 0.30 // etc ...
Pr_4(Heads) = 0.40
Pr_5(Heads) = 0.50 // fair coin
Pr_6(Heads) = 0.60
Pr_7(Heads) = 0.70
Pr_8(Heads) = 0.80
Pr_9(Heads) = 0.90
Pr_10(Heads) = 1.00 // this coin has heads on both sides
Now, let's compute the chance some observed data, i.e. five heads or {H,H,H,H,H}, was
obtained by flipping each coin all by itself:
Coin 0: Pr_0({H,H,H,H,H}) = (.00)^5 = 0.00 // read this as, the probability of getting 5 heads from a coin with zero probability of getting any head is zero.
Coin 1: Pr_1({H,H,H,H,H}} = (.10)^5 = .00001 // read this as, the probability of getting 5 heads from a coin with .10 probability of heads is... well, really small.
Coin 2: Pr_2({H,H,H,H,H}} = (.20)^5 = .00032 // and so on ...
Coin 3: Pr_3({H,H,H,H,H}} = (.30)^5 = .00243
Coin 4: Pr_4({H,H,H,H,H}} = (.40)^5 = .01024
Coin 5: Pr_5({H,H,H,H,H}} = (.50)^5 = .03125 //the probability of getting 5 heads from a fair coin is a bit over 3%
Coin 6: Pr_6({H,H,H,H,H}} = (.60)^5 = .07776
Coin 7: Pr_7({H,H,H,H,H}} = (.70)^5 = .16807
Coin 8: Pr_8({H,H,H,H,H}} = (.80)^5 = .32765
Coin 9: Pr_9({H,H,H,H,H}} = (.90)^5 = .59049
Coin 10: Pr_10({H,H,H,H,H}} = (1.0)^5 = 1.00000
Ok. Ok. Here we go with some interpretation!! First, please notice the above
computation does not yield a probability distribution. BUT what the frack (like
from the movie risky business when tom cruise's dad told him at the end of the movie),
let's simply normalize these values resulting from the coin flip calculations (above)
and say it is a Distribution... (Ok, I'm not really going to normalize them, it takes
too long.) But, this is why:
New Question 1:
I put all 11 of these coins into a dice bag. I pull out one at random and flip it
five times. I get five Heads {H,H,H,H,H}. Which coin would you guess I have picked??
Would you bet the next flip is a Tail??.
Ans: Coin 10 has the highest computed chance of showing 5 heads in a row. I would
not bet the next flip is a Tail.
New Question 2:
Let's do it again. I put the first coin back, and draw another one. And this time I
flip {H,H,T,H,T}. (You have to recompute all the numbers, like above, and normalize.)
Which coin do you think I randomly selected from the bag??
--> Ok (again I'm too lazy to actually normalize) but looking at the numbers we can see
it's most likely Coin 6! With this second set of observed coin flips {H,H,T,H,T}, I
compute Coin 6 was most likely to generate the observed sequence of flips.
And so, would you bet the next flip is a Tail? No!! Because Pr_6(Tail) = 0.40 (40%), and I chose a coin at random.
Without fist specifying, or choosing, a world or a coin, before asking the question, one
has to do calculations over *all possible worlds* and then pick the most-likely world (smart people
say model and arrogant people say hypothesis) to have generated the observed data. The other way around,
specifying the model first (the coin) and then asking the question is waaay too easy -- you
stop getting questions like that in High School. (And you never get questions like that in Science.
Because we don't know the world first, we discover it after the experiment is done. And even then,
all we really do is pick the best, current model of the world.)
** This is why you're calculation above is not a full answer, because you have
a priori locked onto only one coin [Pr(Head)=.50], and there was no reason to do that. <-- that was the trick. Nay, the trap!! **
Finally, you have to have atleast two models and then pick one. Even then all you can really do is
choose between them based upon current data; as more data rolls in, you may switch to a
different model. Also, Probability is not a measure of confidence, it's a measure of luck.
But you will see it basterdized that way all the time.
I want to be in a fast moving spaceship with the velocity to exit the solar system.
Then, when advanced aliens find my frozen corpse they can fiddle with my DNA,
and bring me back to life. song = theme
Ok, we have the answer -> That is not a cross-guard. Those two small,
side beams are just flames venting energy sideways and can't actually
stop anything. What we see is a poorly made light saber by an upstart
Sith, and the blade is venting energy to maintain stability, because the
Sith isn't skilled enough to make a "good" light saber. What looks like
a cross-guard is just a chimney of sorts venting energy sideways harmlessly.