Probability: Elennsar MUST Read

[Username17]

OK first things first. This is not a fucking discussion. Probability is math, and like most forms of math it has answers which are right and it has answers which are wrong. This is not religion or politics where there is any room for your feelings or any of that garbage. If you don't understand this shit, it's because you don't understand it. Maybe you should think about it some more, maybe you have a learning disability. But there's nothing wrong with the theorems being discussed here.

Elennsar, this is your last chance to not get put on permanent ignore. Wrap your mind around this or GTFO.

Iterative Probability: Sets

OK, we're going to talk about these things in terms of coin flips because it is easy. It is important to note that the chances of something happening are always 1. That is while you can land a coin heads half the time and tails half the time, the chance of landing heads or tails is 100%. When you have disjoint events like that, the probabilities do in fact add. But when you're talking about multiple events happening in series, the action tends to be multiplication. Since probability is set in terms of fractions, when you multiply things together they become more unlikely. Getting a head is something that happens 1/2 the time. Getting a head on both flips in your set is something that happens (1/2)*(1/2) = 1/4 of the time.

If you have a situation where you are looking for the chances of one thing happening or (inclusive) another thing happening, it is usually best to find the chances of neither event occurring and subtracting that chance from 1. Remember that chances of any of the set of all possible events happening are always going to add up to 1 because something happens all the time. So your chance of getting at least 1 Head in your set of two flips is easiest found by subtracting your chance of getting Tails on both flips, and then subtracting from 1. So 1 - (1/2) * (1/2) = 3/4. 75% of the time your set will have at least one Head.

The other thing that is important is that our set can be defined however we want to define it. It seriously doesn't matter. Any two flips that haven't been flipped yet have a 1/4 chance of getting two Heads, a 3/4 chance of getting at least 1 Head, and a 1/4 chance of getting 2 Tails. And that adds up to more than 1 because you can't add the chances of getting 2 Heads to the chance of getting at least 1 Head because the chance of getting at least 1 Head already includes the chance of getting both Heads. You can plot those chances out manually by writing down all four combinations (events with at least one Head in them are bolded):

• HH
  HT
  TH
  TT

But the really really important thing about those two coin flips is that literally the only thing that matters about them is that we defined those two as the ones we cared about. And we defined them that way before we flipped them. They aren't the first cons we've ever flipped, nor will they be the last, they just happen to be the ones we're checking.

Imagine for the moment that you flip a coin twenty times in a row. If for whatever reason we are concerned about the 3rd and 15th flip, their chances of both coming up Heads are one in four. If for whatever reason we're concerned about the 10th and 20th, the chances of them both coming up Heads are still one in four. And here's the really really important part: if you define your set by some variable criteria, the chances are still one in four. So if you check the first two flips after someone burps, the chances of them both being Heads are 1 in 4. And the first 2 flips after you get a Head, the chances are still 1 in 4. Yes, even tough that's 3 Heads in a row, the chances are 1 in 4, because only the last 2 of them are actually in the set. You could have gotten a Head on your first flip and then started counting to see if you got 2 Heads in a row, or you could have flipped Tails 16 times in a row before you got a Head and started counting. It doesn't make any difference. That first Head is just a sign post, the set whose probability you are calculating is still just 2 flips.

Selection Bias

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, and the one you got is just the one you got. Every coin you have already flipped no longer has a 50% chance of landing Heads and a similar chance of landing Tails, it has a 100% chance of having landed whichever way it happened to come up.

Let's take the ultimate game of Selection Bias: Let's Make a Deal. Here's how it works: Monty Haul shows you three doors, behind one of which is a prize. You pick one of the doors, (let's say #2 for this example). Then Monty Haul opens one of the other doors (either #1 or #3 in this example) and shows you an area with no prize. Now you are offered the chance to stay on the door you originally picked, or switch to the remaining door.

As it happens, you should switch to the remaining door because of selection bias. The door you originally picked had only a 1 in 3 chance of having the prize. And it still does because nothing has happened to that door to change it. The door Monty showed you the contents of is not random. It's always going to be empty. Which means that if the Prize was behind door #3, he'll open door #1. If the prize is behind door #1, he'll open door #3. So if the prize is behind Door #1 or #3, you'll get the prize if you switch doors. If it was behind door #2, you'll get it if you stay with your original selection. So even though you only have two choices, your chances are not 50/50 because there are additional non-random elements. Your chances are nonetheless calculable, because the effects of the non-random elements are known. The odds are 33% if you stay with your original selection, and 67% if you switch.

How This Affects Game Design

Events can be calculated separately or together. Your chances of making an Appraise flip can be isolated from all your other coin flips simply by announcing that you are doing so. Your chance of flipping Heads on your first Appraise flip is 50%. We can say that with complete confidence even if we don't know how many flips will happen before or after it and don't know what the previous flips actually are.

The set, in that case, is simply one flip. And it has on its own a 50/50 shot of coming up Heads or Tails. If we link it into a set with another flip, then our set is two flips long, and follows the rules for two flip sets above. And no, it doesn't make any difference what the specific criteria of us having two flips in our set are. They could be the first two Attack flips, the first two enemy attack flips, the third appraise flip and the second jump flip. Whatever. Indeed, it could even be something incredibly arcane like the first two flips that meet the criteria of being the first flip after an an enemy has flipped three Heads in a row during an attack cycle. It doesn't matter. Two flips, however they are defined ahead of time, have a 1 in four chance of coming up Heads. And a 3/4 chance of coming up with at least one Head.

-Username17

[Elennsar]

All very nice.

What are the chances that if you die from drinking poisoned wine on a head that you will die of drinking poisoned wine when you may or may not drink the poisoned wine to begin with?

That is the relevant problem to not dying at the banquet as much as the probability of getting a tail (a loss).

[Bigode]

But when you're talking about multiple events happening in series, the action tends to be multiplication. Since probability is set in terms of fractions, when you multiply things together they become more likely. Getting a head is something that happens 1/2 the time. Getting a head on both flips in your set is something that happens (1/2)*(1/2) = 1/4 of the time.

I suppose it should read "less".

If you have a situation where you are looking for the chances of one thing happening or (inclusive) another thing happening, it is usually best to find the chances of neither event occurring and subtracting that chance from 1.

Might wanna explain that "inclusive or" means "any of just one, just other, or both".

[Username17]

It is the chance of drinking poison wine (which is, in fact, some number) times the chance of flipping a head to die. Really, it's that simple.

-Username17

[Elennsar]

The problem is that presumably you choose to drink from the glass or you don't.

At least in my experience, you don't roll to determine if you drink from the goblet in question.

Maybe you'll drink from the first goblet put in front of you. Maybe you'll suspect (or detect) that it is poisoned. Maybe you don't drink wine at all.

Whatever. Not all of these things involve any rolls or flips.

Having the probability that you'll die if a tail kills you does not tell me whether or not you'll be flipping the damn coin to begin with.

[Username17]

Any event which may or may not happen has a chance of happening.

If you do not accept that tautology, I will permanently ignore you.

-Username17

[Elennsar]

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."

[Username17]

OK, I've given you a chance. Now I have added you to the ignore list.

-Username17

[Elennsar]

And since you don't care one whit for what I'm saying because it doesn't say "Yes your math perfectly predicts what people will do, too", I'm not sure there's a noticable difference.

[norms29]

now Frank, let's not be hasty here, you seem to be getting through to him.
Past errors aside, the only thing he seems to be missing here is how we devise meaningful probabilistic statements about non random things.

While I'm reluctant to say so, he seems to be getting the prob\stat portion now. Now you need to go into... Gametheroy, I think.



ih and Elennsar, Off the top of my head I'd say that gamedesign doesn't require us to predict, or even forecast probabilisticly, the player's choices, our purpose in creating game mechanics is to determine what the results of a given choice should be after the choice has been made.

[Elennsar]

My point is, calculating out ahead of time how long a campaign will last unless choices wind up as really simple and limited is not entirely realistic.

You can calculate that IF a given event happens, THEN something will happen, but if that event is never triggered (which is not determined by the dice or the coins or whatever), then people may survive longer (or not as long, depending on the event and the alternate action).

[norms29]

you could, on an abstract level, pick out an assumed level of player competence, and use that to assign probabilities to basic decisions

[Elennsar]

A player who makes the right decision 80% of the time might make the wrong decision the times it could get them killed, but 80% of the time they look great.

That's the problem.

So 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. I'm reasonably confident that I'll make a good decision most of the time (over two thirds of the time?) in Heroes of Migth and Magic. I'm not confident that the times I make a bad decision will be (or won't be) the times it costs me big time - probably not, but I can't begin to guess.

Note: Edited for clarity, as the original words didn't mean what I intended.

[norms29]

I can find no way to parse that sentence wherein those numbers add up properly.

[Elennsar]

That's because the sentence came out jumbled. View the post with edits, hope that came out clearly.

[Prak]

I'd like to say that I, at least, found this helpful Probability and statistics were not my strong point in school.

Also, Elenssar, We don't need to predict whether or not someone will drink the poisoned wine. To be honest, someone will always drink the wine, but that's not probability, that's people, and, if anything, psychology.

What we need to predict is the likely hood someone will survive drinking the poisoned wine. If it requires three heads in a row to make you save vrs. poison then a player has a 1/8 chance (if my math's right) to survive drinking the poisoned wine.

Predicting actions is a fool's errand, predicting outcomes, that can be done.

[norms29]

unless we identifiy some other factor influencing when they make good/bad decisions, we can treat it as any other iterative probability

80% (to use your example) * (probability of out comes for "right" decision) and 20%*outcomes for wrong decision(s).

the tricky part, identifiying what the odds of right decision vs. wrong, is something you need to pick an initial assumption for, the values in the final product are determined by complexity of the rules.

[Elennsar]

I think the statistics are fine, as statistics. But:

I won't drink the wine - as said, I don't drink.

Bob will drink the wine. Inevitably. Or close enough as to make no difference. (whether or not anything is inevitable is philosophy, not math)

Bob has a 12.5% chance of surviving.

Predicting "whether or not the PCs will survive" when they may or may not touch the stuff doesn't give us any real idea on how long the PCs will survive in actual play, because they may not do the actions the predictions were based on.

Norms: A nice number. Now, what the hell do we do with it?

Using this to predict the outcome of a campaign is only as good as how good the guesses on what actions will be taken and what events will be triggered is.

[Prak]

The real problem here is that you're trying to use this in a vaguely ambiguous circumstance. AFA I can tell, statistics are a lot more helpful in combat situations, where you can know what decisions are going to be made than at the thieve's guild gala where 1 in 4 glasses of wine contain dragon bile.

[Elennsar]

"the whole of a campaign" is a lot more of an ambiguous circumstance in the sense you just used the term than a perfectly predictable one.

[Prak]

true, but I find it somewhat telling that class acid tests consist of leagues of monsters and traps, and not a single poisoning attempt.

You could probably do complex analyses of people to figure some probability of a person drinking the poisoned wine, but... I don't have that kind of patience or know how, nor am I that inclined to analyze every single damned possibility of the whole of the campaign. Really talk of a character's longevity is based more on experience than statistics, it seems.

[violence in the media]

Elennsar, for determining the probability of player actions, you just make an initial assumption and revise it as you get more information.

You might say that a player is 50% likely to drink the wine as an initial starting point. Then, through analysis and play testing, you discover that players are more suspicious of things like glasses of wine and, because there's no reason for them to drink the wine, some of them are way more likely to drink it then others.

So you break your data out into two groups now. You have the Gamers, who will rarely drink the wine and the Role Players who will usually drink the wine. The Gamers drink the wine only 15% of the time, whereas the Role Players drink the wine 75% of the time.

And so on.

As norms29 said though, game design doesn't concern itself that much with people's choices. You don't really have to care about how the character gets poisoned, just that the system works the way you want it to once they do. Unless you expand to include it whether or not they drink it in the first place, the math of surviving the poisoned goblet is only focused on what happens after the initial condition (drinking the wine) is met.

[Elennsar]

You don't really have to care about how the character gets poisoned, just that the system works the way you want it to once they do. Unless you expand to include it whether or not they drink it in the first place, the math of surviving the poisoned goblet is only focused on what happens after the initial condition (drinking the wine) is met.

Personally, I'm very much concerned as to how they get poisoned, because unless and until that condition is met, that you can die from it is unimportant.

And that's why I keep saying over and over again that it depends on what the character does.

Having a 50% chance of dying if you run into a giant lion when you have to deliberately seek out giant lions to have any chance of finding one may make you much more likely to die or may not even matter because the event never comes up.

[Judging__Eagle]

I hope that Frank will take the time out of his busy schedule studying to be a doctor to show me how to learn things that I don't seem to quite understand. Hopefully he'll never think that I need a personalized lesson, and if I do, I hope that I'll be able to wrap my head around it.

This was an interesting read, I sort of knew how this stuff worked (odds are the same no matter how often you do something, unless you change the way that odds are being calculated; grinding monsters for specific items in games like Ragnarok Online taught me that, each mob has a set chance and that chance never changes, so you can just as easily get a rare drop off a monster on your first two kills as in your first two hundred; or more like two thousand; the percent is always 0.1% for X item (rare item) and 110% for Y item (npc-trash)).

However, having it explained so succinctly allows me to direct other people towards this post when explaining probability to them.

[zeruslord]

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.