Probability: Elennsar MUST Read

[MartinHarper]

Someone who may or may not drink a given glass of wine has a probability of drinking that glass of wine. That probability is some number between 0 and 1 (or between 0% and 100%). It is not a probability that you or I know, but it still exists.

Similarly, I was born at some point in the past. I have an age, which is the time that has passed since I was born. You do not know that age, but that does not mean that I am ageless. Equally, I do not know your age, but you are not ageless either.

What are the odds that someone making a bad decision will be at the time a bad decision could kill them? Damned if I know.

If you read and understand Frank's first post, you'll find out.

[Elennsar]

If you are desperate and patient enough, you can empirically determine the probabilities of a player doing something by sitting every person who plays your game down and running them through a scenario, then dividing the number of people who performed each action by the number of people who could perform each action.

That sounds like an ineffective use of time, to say the least.

If you read and understand Frank's first post, you'll find out.

And if you read and understand my objection to insisting that trying to predict whether or not the characters will die of poisoned wine, you'll see my point.

[violence in the media]

If you are desperate and patient enough, you can empirically determine the probabilities of a player doing something by sitting every person who plays your game down and running them through a scenario, then dividing the number of people who performed each action by the number of people who could perform each action.

That sounds like an ineffective use of time, to say the least.

Then you have to make a best guess and move on with your life. No one cares if your assessment of the probability of drinking wine is correct, just that it's the stated assumption you're working on and that you'll revise it in the face of evidence.

And if you read and understand my objection to insisting that trying to predict whether or not the characters will die of poisoned wine, you'll see my point.

Your point is irrelevant and meaningless. Why? Because you're throwing your hands up in the air and insisting that because the answer is unknown, it is also unknowable.

[Elennsar]

No one cares if your assessment of the probability of drinking wine is correct, just that it's the stated assumption you're working on and that you'll revise it in the face of evidence.

Then what purpose does the assumption serve? To come up with a number you have no intent of achieving OR avoiding so that when if it reaches that you have no idea if that was a good thing?

Your point is irrelevant and meaningless. Why? Because you're throwing your hands up in the air and insisting that because the answer is unknown, it is also unknowable.

No, I am insisting that it is impossible to state "It will happen, and therefore we know exactly what will happen from there."

There is a chance greater than zero of me watching Ben and Me on Youtube before Thursday. How big a chance? I don't know.

Me telling you that "There's some chance" would be actually useful information. Me telling you there's one chance in eight would be giving a "concrete" number that has no real use or meaning.

[Leress]

And if you read and understand my objection to insisting that trying to predict whether or not the characters will die of poisoned wine, you'll see my point.

I see someone doesn't know how to apply an example.

Me telling you that "There's some chance" would be actually useful information. Me telling you there's one chance in eight would be giving a "concrete" number that has no real use or meaning.

How is being vague more useful then being more specific?


You seem to be mixing up when probability is to be used. You don't use it on the players decision, you use it on the possible outcome of said decision.

[Elennsar]

I see someone doesn't know how to apply an example.

I see someone is missing the problem with using the probability statistics to determine how long characters will survive a campaign.

How is being vague more useful then being more specific?

Because there is an excellent chance that "1 in 8" is complete garbage. So me saying "1 in 8" instead of "X in Y" doesn't actually let you predict whether or not I'm going to have youtube freeze up on me, so predicting what a hypothetical person who would do it would do doesn't tell me anything I need to or want to know.

You seem to be mixing up when probability is to be used. You don't use it on the players decision, you use it on the possible outcome of said decision.

You seem to be ignoring my problem with claiming "and your campaign will last this long if this is the case" after coming up with a bit of data that may or may not be true in ANY campaign and crunching numbers.

If all you want is "You survive 50% of the time if X happens", then pick your own numbers and run them until your calculators wear out. Since someone might not choose to do X, that doesn't determine how long you'll live in actual play at all.

[Judging__Eagle]

I see someone is missing the problem with using the probability statistics to determine how long characters will survive a campaign.

Just look at the odds of surviving each encounter, then the odds of surviving every encounter. I mean, you can do that, right? You have an inkling as to how hard it is to survive each encounter, right?

Now, on an infinite timeline, multiply all of your odds by 0. That's the odds of surviving an infinite timeline.

On an infinite timeline, everyone dies. Even god-emperors. As in, the original god-emperor, Leto Atredies II (predating the inital release of any 40k product by 7 years).

Really, it's sort of a wonky inverse proportion graph, characters that never see play always live forever; characters that play forever will invariably be rendered un-playable at some point in time.

Unplayable does not mean dead. When your character gives up everything and chooses to permanently polymorphed into wolf, your character is as unplayable as one that was killed in a TPK, or on their own and killed (with no allies knowing that they died).

[Elennsar]

On a short of infinite timeline, you still have a variety of things that influence your chances and influence whether or not you even deal with your chances of dying at any given point.

As for surviving every encounter, the problem is that hero points render that different than the probability of the math - and you may or may not have them when you need them, so its not even "with them, you have these odds, so your odds of surviving the campaign are ___"

So if you have a 10% chance of dying without spending a hero point, then you can calculate the odds of surviving 40 encounters. What are the odds of you taking part in all 40 encounters (injuried or not)? Good question. What are the odds you'll have a hero point whenever you fail your "vs. death roll"? Good question.

Bloody complicated business.

[violence in the media]

No one cares if your assessment of the probability of drinking wine is correct, just that it's the stated assumption you're working on and that you'll revise it in the face of evidence.

Then what purpose does the assumption serve? To come up with a number you have no intent of achieving OR avoiding so that when if it reaches that you have no idea if that was a good thing?

The assumption gives you a starting point to work from. Lets say that you feel a poisoned goblet is almost always fatal for anyone who drinks from it (they survive on a 17+, for instance). Because of this, you want to establish what proportion of players are going to drink from the goblet and then have their characters die.

You don't view this as a crystal ball prediction of Elennsar the White Knight's future action, you use it to get a general sense of how many people drink from the goblet out of the population of people playing your game. Some tables will have no one drink the poison, some tables will TPK because of it. Knight Elennsar may never drink from the goblet, whereas Knight VitM might always do so.

So, you take your guess that X% of people will drink the poison, Y% will die from it, and that equals the assumption that Z% of the player base that is killed by poison wine when confronted with it. So, you follow up by checking your message boards, or collecting table data after the convention, and you compare this to your initial assumption. You can then change things in the face of this new data. Maybe too many people drink the wine, in which case you decide to make it less fatal. Or maybe far fewer people drink the wine, and you decide to make it a death sentence.

Studies similar to this are done in sociology and economics and manufacturing all the time.

Your point is irrelevant and meaningless. Why? Because you're throwing your hands up in the air and insisting that because the answer is unknown, it is also unknowable.

No, I am insisting that it is impossible to state "It will happen, and therefore we know exactly what will happen from there."

Again, we don't care about predicting exactly what Elennsar is going to do, we care that 32.78% of the player base is going to get killed by poisoned wine. If that's an acceptible number, then we move on.

There is a chance greater than zero of me watching Ben and Me on Youtube before Thursday. How big a chance? I don't know.

Me telling you that "There's some chance" would be actually useful information. Me telling you there's one chance in eight would be giving a "concrete" number that has no real use or meaning.

If we took data from a year of you watching (or not) Ben and Me weekly on Youtube and identified some influencing factors in your decision, then yes, we could come up with a concrete number that accurately indicates how likely you are to watch it in a given week.

[Elennsar]

So, you take your guess that X% of people will drink the poison, Y% will die from it, and that equals the assumption that Z% of the player base that is killed by poison wine when confronted with it. So, you follow up by checking your message boards, or collecting table data after the convention, and you compare this to your initial assumption. You can then change things in the face of this new data. Maybe too many people drink the wine, in which case you decide to make it less fatal. Or maybe far fewer people drink the wine, and you decide to make it a death sentence.

Studies similar to this are done in sociology and economics and manufacturing all the time.

The problem is that I don't have the first idea how many people will drink the poison - and making up a random number just to appease your insistance on having one wouldn't help at all.

Again, we don't care about predicting exactly what Elennsar is going to do, we care that 32.78% of the player base is going to get killed by poisoned wine. If that's an acceptible number, then we move on.

And again, you need to have some idea what the player base is going to do before knowing if 32.78% get killed.

If we took data from a year of you watching (or not) Ben and Me weekly on Youtube and identified some influencing factors in your decision, then yes, we could come up with a concrete number that accurately indicates how likely you are to watch it in a given week.

Right, but without that actual data, "1 in 8" is a wild guess that might be anywhere from exactly right to to totally false.

In the case of the wine, I don't know what would be a good number - if I had an idea, I'd submit, but I don't.

Maybe it would be good if a third of the players die of poisoned wine. Maybe that's a bit too high. Maybe its a bit too low. Maybe its a lot too high or too low.

[violence in the media]

Elennsar, the problem is that you are refusing to start working towards a solution and won't even acknowledge that a solution is possible. Not just in the poisoned wine scenario, but in any scenario for the entire game.

It does not have to be right on the first guess! A wild guess, even if it winds up being totally wrong, at least gives you somewhere to start.

Going back to other examples people have talked about, like combat, and you cannot say what percentage of the time you want one knight to hit another knight with his sword. And that's something that is guaranteed to happen at some point in any campaign! It doesn't even involve any choice on the part of the player. You, the DM, simply declare that the knight draws his sword and attacks Sir MartinHarper, and it happens.

What percentage of the time does that initial attack hit?

[shau]

While we are teaching probability, can someone better explain the Monty Hall problem to me. I first heard it as an undergrad and I did not understand it then and I do not understand it now. It seems like you are ultimately given a choice between two options (door 1 and door 2, door 3 has been eliminated) and somehow switching to door 2 from an original selection of door 1 is advantageous, but I have never been able to grasp how.

[zeruslord]

When you pick door 1, you know that there is a 33% chance of the prize being behind that door. That does not change. This means that there is a 66% chance of it being behind one of the other two doors. This also does not change. Monty then reveals which of those two doors does not contain the prize in the 66% where one of them does. That means that there is a 66% chance of the prize being behind the other door you did not pick.

[MartinHarper]

While we are teaching probability, can someone better explain the Monty Hall problem to me.

I think it's often badly explained. Here's my attempt.

There is a game show, whereby there are three doors presented to the contestant. One of them has a car behind it, the other two have goats. The game show always works the same way, every week, and it works as follows:
1. The contestant picks a door, but does not open it.
2. If the contestant picks the car door, then one of the two goat doors swings open, at random.
3. If the contestant picks one of the goat doors, then the other goat door swings open.
4. The contestant then picks one of the two closed doors: either the door she originally picked, or the door that she did not originally pick.
5. The contestant opens the door from her second choice, and wins either a car, or a goat.

The math works out that the door that was initially picked has a 1/3 chance of hiding the car, while the other door has a 2/3 chance. This can be confirmed by experiment.

[Username17]

Sure. The key is that Monty gives you non-random information in the middle of the problem.

There are originally three states:

• Prize is behind Door #1
• Prize is behind Door #2
• Prize is behind Door #3

And you pick a door. Let's say you pick Door #1.

Then Monty opens a door that the prize is not behind that isn't the door you originally chose. He opens door #3.

Now you have the choice of Door #1 or Door #2. But the thing is that Monty just gave you non-random information about Door #2. What he told you was that if the Prize was behind Door #2 or Door #3, then it was behind Door #2.

So in essence, of the original 3 possibilities, 2 of them are equivalent to switching.

---

Another way to look at it is the procedure map from the point of view of Monty. Let's say you pick Door number 1. Monty looks at the prize and selects the door he is going to open based on where its real location is:

• Prize is behind Door #1: Monty shows you a random Door, 50% #2, 50% #3.
• Prize is behind Door #2: Monty opens Door #3 100% of the time.
• Prize is behind Door #3: Monty opens Door #2 100% of the time.

So when you see that Door #3 is opened, you know that there was a full chance that it would be opened if #2 was the prize, and only a half chance that it would be opened if #1 held the prize. So 2/3 of the time, you'll get the prize by switching to #2.

-Username17

[RandomCasualty2]

As for surviving every encounter, the problem is that hero points render that different than the probability of the math - and you may or may not have them when you need them, so its not even "with them, you have these odds, so your odds of surviving the campaign are ___"

So if you have a 10% chance of dying without spending a hero point, then you can calculate the odds of surviving 40 encounters. What are the odds of you taking part in all 40 encounters (injuried or not)? Good question. What are the odds you'll have a hero point whenever you fail your "vs. death roll"? Good question.

Bloody complicated business.

Yeah, hero points make the math harder, but they don't invalidate it. Adding hero points changes the probability. That's all it does.

If you get a reroll somewhere in the combat, that just moves the probability more in your favor.

It doesn't perform some weird magic that renders the laws of probability invalid as you seem to think.

Just because the math is difficult to calculate doesn't mean a percentage doesn't exist. That's a fallacy of bad game designers.

[Psychic Robot]

To elaborate on what others have said: it's easier if you imagine it the game with more doors. Let's say you have 100 doors. There are goats behind 99 of them, and there's a car behind 1 of them. You choose one of them. You have 1/100 chance of picking the right door, correct? Well, Monty Hall comes down and opens up 98 of the other doors, leaving you with your door and a mystery door. He offers you the chance to swap doors. What do you do?

Well, you switch, since there was a 1/100 chance that you picked the right door on the first try. That's the important part--think about it like that, and the weird math becomes easier to understand. There's a 100/100 percent chance of one of the doors has a car behind it. If your choice has only a 1/100 chance of being right, you just subtract the 1/100 from the 100/100--and, voila, there's a 99/100 chance that the other door is the door with the car.

You can also examine this in terms of outcomes. Let's say that you choose Door #1.

If Door #1 has the car behind it, Monty will open Door #2 or #3. If you switch to either one, you will lose. (A loss from switching that occurs 1/3 of the time.)

If Door #2 has the car behind it, Monty will open Door #3. If you switch, you will win. If you don't switch, you will lose. (A win from switching that occurs 1/3 of the time.)

If Door #3 has the car behind it, Monty will open Door #2. If you switch, you will win. If you don't switch, you will lose. (A win from switching that occurs 1/3 of the time.)

In the scenarios, you will end up with a 2/3 chance of winning from switching.

[RandomCasualty2]

Basically the door puzzle is this.

You have two choices.

The first door you pick is 1 out of 3. When offered to choose, you are in fact offered a choice of your first door (1 out of 3) or the best of the remaining two doors (2 out of 3).

[the_taken]

A chance, yes. A chance you can always put in and get a percentage of, no.

I don't drink, for instance. That you can put in.

Someone who drinks every other day, you can put in.

Someone who drinks "sometimes", not until sometimes becomes a "Yes, this time." or "No, not this time."

Lost me right here. Can you explain this, plz?

[Josh_Kablack]

The chances of things having turned out the way they did is 100%. Always. No matter how weird the data set seems. You were always going to end up with a result,

This is tautologically true.

The issue comes with applying it correctly, especially to sets of open-ended responses, such as human behavior.

Since I currently work in data collection I see a lot of folks set up surveys with "Yes" or "No" answers - and the PhDs who design such questions like to think that those are the only two answers - which is a woefully incomplete response set.

When dealing with real people, there are are least five possible answers to any such question. Most folks will give the "Yes" or "No" response you want - some folks will refuse to answer ("That question is too personal"); some folks who cannot understand the question ('No habla anglais"; "Spreak up sonny, I'm deaf") and folks who understand the question but insist on an alternate response ("That's not clear cut, I can't say yes or now, instead I say sometimes"; "Oh god, we're having an earthquake *click*").

Thus when dealing with human behavior, it's still true that something always happens - but that particular something may not always fall within the preconceived response set. If a response set and the subsequent data analysis is not set up to handle these possibilities you get the weird-looking results where the total of all responses does not add up to 100%.

[opera]

In the case of the wine, I don't know what would be a good number - if I had an idea, I'd submit, but I don't.

Perhaps this would all make more sense if you didn't think about the probabilities involved as single numbers/frequencies, but thought about the issue in terms of a probability distribution.

For example, you've said that you aren't comfortable coming up with a specific frequency for someone drinking poisoned wine, but your statement actually describes (informally) a very specific probability distribution - a uniform distribution, often used in certain areas of statistics and decision theory to represent complete ignorance about the likelihoods of expected outcomes. If you wanted, you could then use that probability distribution, combined with the probability distribution associated with dying from poisoned wine conditional on drinking the wine, and come up with a distribution representing the overall probability of a character dying from poisoned wine.

Of course, a uniform distribution is probably not a very good guess.... if you sat down and thought about it, you might decide that there are probably going to be fewer people who never drink the wine or always drink the wine than people who sometimes drink the wine. This suggests that you might be more comfortable with positing something like a normal distribution with very high variance (e.g., a relatively "flat" distribution, like a squished hill). Making this change, of course, would alter the expected likelihood of a given character dying from poisoning.

Remember, it's perfectly all right to say you don't agree with the specific frequencies or probability distributions suggested by other people, if you can make a case that they are misrepresenting the sample space (i.e., the space representing the range of possible outcomes) or the actual likelihoods of certain events; what you can't say is that such events can't be modeled mathematically according to well-known rules.

[virgil]

what you can't say is that such events can't be modeled mathematically according to well-known rules.

Oh, you 'can' say it, and I'm sure Elennsar will do his damndest to do so. Just like I 'can' say that I'm made out of brie.

[CatharzGodfoot]

Some philosophy might help...

Where Do Probabilities Come From?

There has been endless debate over the source and status of probability numbers.
The frequentist position is that the numbers can come only from experiments: if
we test 100 people and find that 10 of them have a cavity, then we can say that
the probability of a cavity is approximately 0.1. In this view, the assertion "the
probability of a cavity is 0.1" means that 0.1 is the fraction that would be observed
in the limit of infinitely many samples. From any finite sample, we can estimate
the true fraction and also calculate how accurate our estimate is likely to be.

The objectivist view is that probabilities are real aspects of the universepropensities
of objects to behave in certain ways-rather than being just descriptions
of an observer's degree of belief. For example, that a fair coin comes up
heads with probability 0.5 is a propensity of the coin itself. In this view, frequentist
measurements are attempts to observe these propensities. Most physicists
agree that quantum phenomena are objectively probabilistic, but uncertainty at the
macroscopic scale--e.g., in coin tossing-usually arises from ignorance of initial
conditions and does not seem consistent with the propensity view.

The subjectivist view describes probabilities as a way of characterizing an
agent's beliefs, rather than as having any external physical significance. This allows
the doctor or analyst to make the numbers u p t o say, "In my opinion, I expect the
probability of a cavity to be about 0.1 ." Several more reliable techniques, such as
the betting systems described on page 474, have also been developed for eliciting
probability assessments from humans.


In the end, even a strict frequentist position involves subjective analysis, so the
difference probably has little practical importance. The reference class problem
illustrates the intrusion of subjectivity. Suppose that a frequentist doctor wants to
know the chances that a patient has a particular disease. The doctor wants to consider
other patients who are similar in important ways-age, symptoms, perhaps
sex-and see what proportion of them had the disease. But if the doctor considered
everything that is known about the patient-weight to the nearest gram, hair
color, mother's maiden name, etc.-the result would be that there are no other patients
who are exactly the same and thus no reference class from which to collect
experimental data. This has been a vexing problem in the philosophy of science.

Laplace's principle of indifference (1816) states that propositions that are
syntactically "symmetric" with respect to the evidence should be accorded equal
probability. Various refinements have been proposed, culminating in the attempt
by Carnap and others to develop a rigorous inductive logic, capable of computing
the correct probability for any proposition from any collection of observations.
Currently, it is believed that no unique inductive logic exists; rather, any such logic
rests on a subjective prior probability distribution whose effect is diminished as
more observations are collected.

[shau]

Thanks everyone. The Monty Hall problem seems like something I can follow the logic on, but still refuse to accept on some level, if that makes any sense. I am probably going to have some people do the experiment with me before I get it.

[Elennsar]

What percentage of the time does that initial attack hit?

Depends on how what you do, and how skilled you are, and how skilled the other guy is.

So somewhere less than (usually) 100% of the time the attack beats the to-hit TN and somewhere less than (usually) 100% the defense defeats it.

How much? A random guess would not help here!

I don't even know if I want the average attack hitting (but that being okay because your armor saves you) or not. Until that decision is made, picking a number doesn't do anything.

It doesn't perform some weird magic that renders the laws of probability invalid as you seem to think.

No, but it does do something where you can go from a 1-10 chance of success based on the dice to, since you have a hero point and a hero point can boost your margin of success (or reduce your margin of failure) by 1 (for instance) - the likelyhood of succeeding because of the dice becomes considerably more complicated.

Lost me right here. Can you explain this, plz?

"Sometimes" is not a number. "Sometimes" means that right up until we know what the number is, is just X.

what you can't say is that such events can't be modeled mathematically according to well-known rules.

I -can- say that unless and until we have some idea how many people drink the wine, knowing how many people survive it is is going to be pure guesswork. And I don't have the first idea how many people will. I don't know whether or not most people should be drinking the wine or not.

So if someone wants to put in something for X, they're welcome to do so. Picking a random number would not produce a useful result until we know what we want it to reach (or what we don't want it to reach, besides presumably total death of everyone, even those who don't drink wine).

Oh, you 'can' say it, and I'm sure Elennsar will do his damndest to do so. Just like I 'can' say that I'm made out of brie.

No, I will do my damnest to point out that being able to model something when we have the numbers and being able to give numbers that mean anything are two different steps.

Obviously, we can just pull "50%" out of our asses, and the fact we don't know if we consider that a reasonable basis to see if it is sufficiently lethal doesn't matter because we don't want to actually do something useful, we want to run numbers for its own sake.

I don't know if you should -want- to use a flurry of blows vs. an equally skilled opponent, or if it is best vs. an inferior one.

When that can be answered, how likely it is to hit assuming equal skill can begin to be answering, but until then, any answer would be "yeah, if we did it that way, but are we planning on doing it that way? Maybe."

So my question is, why do you contiuously ignore the fact that until you know whether or not you want 50% of the players playing to die, figuring out what numbers drinking and what numbers from there failing it would take for that to happen is unproductive?