# Worlds of Design: Always Tell Me the Odds

If GMs (and game designers, and gamers) understand “the odds” they will be able to make better choices and understand why some things happen in their games - and some don’t.

Picture courtesy of Pixabay.
Never tell me the odds!
--Han Solo (Star Wars)​

Most people don't understand odds and randomness in the most simple dimensions, especially when you're talking about dynamic odds.
--Keith S. Whyte. Executive Director. National Council on Problem Gambling​

We often hear about “the percentages” and “the odds” in sports. For example, the odds for the home team winning (regular season: NBA 59.9%, NFL 57.1, NHL 55.1, MLB 54.0, MLS soccer (where there are draws) home win ratio of 49.4 percent over a 15 year period, compared to just 26.5 percent away wins). Though game design does not require higher math, game designers need to know simple arithmetic and probability. There are some odds we can talk about in RPGs, as well, and about how people react to those odds.

The notion that it can be a "fair fight" in an RPG? 50/50? Nope.

How much is a fight biased toward the adventurers? Let’s consider the NCAA Basketball tournament. Let’s say that a team is so good, it can win 90% of its games against the better teams, the ones in the tournament. This is unlikely: how many teams have a season record as good as 27-3 (90%) though they’re playing weak as well as strong teams? When you’re playing the stronger teams, 90% is quite unlikely. But let’s use that anyway.

So what are the chances of winning the tournament (six games in a row) even with that 90% (beyond-likelihood) capability?

 90%​ win 1 in a row​ 81.00%​ win 2 in a row​ 72.90%​ win 3 in a row​ 65.61%​ win 4 in a row​ 59.05%​ win 5 in a row​ 53.14%​ win 6 in a row​

Even that most unlikely team that can win 90% of games against tournament-quality opposition, only has a 53.14% chance of winning the tournament. Even a team with a 99% win likelihood wins the six-game tournament only 94.15% of the time (“fail on a roll of 1 on d20").

(How is this calculated? You multiply, you don't add. So to win three games in a row, it’s 90% times 90% times 90%.)

This is why the “best team” often fails to win the tournament. This is why some pro sports play seven-game playoff series, in the hope that luck “evens out” and the better team will win.

Translate This into RPGs

Extrapolate that into RPG sessions with perhaps one big battle per session, or maybe more! Practically speaking, either you need really astute players willing to run away from almost any encounter, in order to avoid taking chances, or you need to arrange a huge bias in favor of the players in a typical encounter. Or they're going to lose and possibly die pretty soon.

Go back to the tournament example. If the players are 90% likely to win, after six encounters there will be around a 47% chance that they will have lost one of those encounters.

The whole notion of RPG combat as "sport", as something that's "fair", is nonsense in light of these calculations.

Playing Styles

Some play for "the rush", for glory, and like Han Solo don't want to know the odds before they do something. If you accommodate them, then the bias in favor of the players must be even greater, or you'll have dead characters in no time. (This brings up the question of "fudging" dice rolls in favor of characters, which I may address another time. Some GMs do it routinely, others never.)

Is it fun to play to survive, to “win”, instead of for glory? Depends on the person. It is for me, when I expand it to include survival for the entire group, not just my character(s). In contrast, in the late 70s I played in a game that was supposed to act as the stimulus for someone to write a story. I tried to do something "heroic". My character got dead.

Many gamers don't understand probability, and so over- (or under-) estimate their chances of success. Some don't understand the scope of the chances. 1 in a thousand vs 1 in a million is a massive difference, but people often don't see it that way. It's yet another case of perception not matching reality.

That's where we get those who don't understand odds, who think that anything (no matter how outlandish) ought to be possible once in 20 (a 20 on a d20) or at worst once in a hundred (100 on percentage dice). No, the chance of most anything happening in a given situation are astronomically against. (And "astronomically" is practically the same as "impossible".)

Recently I talked with a gamer who is very skeptical of probabilities, but doesn't understand them. He thought it was terribly unlikely that a player could roll five dice in a row and get at least a 4 on every roll. The chances, 50% to the fifth power, amount to better than 3%. For some reason he thought that rolling the dice successively rather than altogether made a difference - nope, what's come before has no bearing on what comes after, in odds. And what about five 1's in a row? That's 16.66% (a 1 on a d6) to the fifth, .000129 or .0129%. One tenth of one percent (one chance in a thousand) is .01%. So slightly better than one chance in a thousand. Rolling seven 1's in a row is about 3.5 chances in a million. Or perhaps more easily, rolling a 1 on every one of six 10-sided dice is a one-in-a-million chance.

To summarize: For designers, fudging the dice (or the quality of the opposition) is inevitable. For players, it helps to understand probabilities in games

Reference: James Ernest (Cheapass Games) - Probability for Game Designers | League of Gamemakers

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### Lewis Pulsipher

Dragon, White Dwarf, Fiend Folio

#### clearstream

##### (He, Him)
Pardon my generalization, but this is a very D&D-type calculation. Where the most common odds are win or die. Other shades of loss are not as commonly seen. (And when survival and win are the same, we often get players conflating their character winning with them winning, which is a closely related but completely separate problem.)
It will be true wherever a character might be played for multiple sessions while there is any non-zero chance of their becoming unplayable as an outcome of the game mechanics.

D&D, Pathfinder, RuneQuest, Earthdawn, Shadowrun, Bushido, et al.

That can be nuanced as you say, of course, but the change in the wager over time is also profound.

#### CapnZapp

##### Legend
So, yeah, a DM is absolutely "bound to offer only mathematically sound options. " A DM who forgets that is just screwing over his players.
Sorry, that's unconstructive and antagonistic to me.

You might find that changing your uncompromising stand makes more DMs willing to have you as a player.

#### Hussar

##### Legend
@Hussar STA playtest had much more concrete advice than the release. It was actually quite nice to have the explicit costs in threat.

The release does have some advice - either a trait, an unpleasant fact, or a change in threat rating... and traits are the really vague part:
1. +1 to some type of action
2. -1 to some type of action
3. Prohibit some kind of action
4. Enable some kind of action
5. use with momentum to establish a fact
6. (rarely) Allows purchase of a talent
This is where things start to fall apart: 3&4 are much stronger than 1, 2, and 5. 6 is just there for racial traits. But they have the same cost: 2 threat, or 1 complication.

Yeah, I've been noodling around on Reddit and there's a number of fixes for this. We've just started with the system, so, still ironing kinks out.

So, yes, it's entirely a fixable issue. 100%. Now we'll have to see if the person running the game sees the issue or not.

#### Hussar

##### Legend
Sorry, that's unconstructive and antagonistic to me.

You might find that changing your uncompromising stand makes more DMs willing to have you as a player.

Heh. Never had a problem with that.

I find the larger problem of DM's being unwilling to compromise and accept that they can just as easily make mistakes as the rest of us makes for awful DM's and bad games. How is it antagonistic to expect the DM's homebrew options not to be a mechanical screw job? How is it compromising to accept that the DM is wrong, flat out wrong, but, refuses to accept that he or she has made a mistake?

Sorry, if you want to sit in the big daddy chair, then you get to wear the big daddy pants and take responsibility for your mistakes. And, frankly, many DM's out there are very, very bad at this sort of math and it shows. All you have to do is look at examples on the forums of how folks would "rule" various things to see that there's a significant number of DM's out there that really should be held to a higher standard than they are.

I'm sorry you think that, "Hey, I expect your homebrew material to not screw me over, and, when I show you that yes, you are indeed screwing me over, can we please change your house rules" is an antagonistic and uncompromising stand, but, well, it's a standard I think all DM's should be holding themselves to. It's certainly the standard I try to hold myself to.

#### clearstream

##### (He, Him)
Not sure we're talking about the same thing.

If I can spend a "build point" (or whatever) on getting +5% to my Diplomacy skill of 45%, or to "Murder With Axe", or whatever, I know what I'm getting. What I'm getting is a Diplomacy skill of 50%. Or, given everything equal, I hit with 10 out of 20 axe-swings instead of 9 out of 20 swings.

If I instead get an extra die to my pool of four dice, what do I get?

To me, the number crunching needed to arrive at "a fifth die increases my odds from 47% to 51.9%", ergo I get 4.9% for my money, is entirely and wholly unwelcome. Why would anyone want to go through this step (in practice you need an online probability calculator)?

Zapp

PS. I mean, apparently the answer is "some don't care", so hey, this man's garbage is your treasure, and so on...
I mean that people can know the odds - 60% say - while not being able to clearly visualise what that should entail. How it should impact on their decision-making. Features like the number of checks in the given timeframe, and what might interact with those checks, matter.

Take your example of hitting on a) 10 out of 20 swings instead of b) 9 out of 20. This is meaningful in a few ways. In a) I expect to hit once for each two swings I make. How should I think about that if I am only making one swing in a combat? What about if I get to make 5 swings? What is the consequence of a swing? What action do I give up to make that swing? What if I have advantage or disadvantage? What about inspiration? What about a flat bonus or penalty? Even keeping things very simple, it is unlikely the player has in mind a probability distribution function that is up to the job of predicting exactly how this plays out. They need mental tools for interpreting the %age. And really, a %age is no more beneficial than any other representation of odds. It give us nice round numbers - so great - but so does d20.

What is the difference between 10/20 and 9/20 going to be experientially? Knowing the %age doesn't give you the whole picture. What I should be thinking is something like - combats are about four rounds (say) and I make a swing per round and it takes three swings to deal enough damage to drop this ogre so perhaps I need to get out of here. The ogre probably only hits me 15% of the time (I might not know what those odds are), but if it does I'm likely to be downed (or I might not know what the damage range is, only that it is large).

Diplomacy is usually used in a contest situation, or to beat a threshold. Without knowing that contest or threshold, +5% doesn't tell you much. Yes, it is better. So was getting the extra die.

OTOH what I think you are saying is that game mechanics shouldn't needlessly obfuscate their performance: that, I can get on-board with.

#### CapnZapp

##### Legend
Heh. Never had a problem with that.
I suspect you don't see the irony in that statement.

Anyway, we're done here.

#### CapnZapp

##### Legend
And really, a %age is no more beneficial than any other representation of odds. It give us nice round numbers - so great - but so does d20.
45% is certainly no more helpful than "9 out of 20".

But I'm not talking about that, I'm contrasting transparent to opaque odds.

Something like "45%" or "get +1 to your d20 roll" is transparent.

Something like "get +1 to your 2d6 roll" is immediately less transparent, since it requires an understanding of bell curves. Same with D&D's advantage "roll two dice, pick the best" (which gives the most benefit when your original success rate was 50%).

And then there's the profoundly opaque resolution systems of dice pool games: How likely are you to "roll five dice, get two successes"? Which is best - "gain an extra die" or "get +1 to all your dice"?

Cheers

#### Fenris-77

##### Small God of the Dozens
Supporter
I feel like we could, as a forum, easily produce a stats primer of some kind that explains some of the basic math behind some of these examples. It doesn't need to be 100% precise, just close enough for people to make better decisions. Maybe a wiki...

#### dragoner

##### KosmicRPG.com
To summarize: For designers, fudging the dice (or the quality of the opposition) is inevitable. For players, it helps to understand probabilities in games

The Traveller Book (1982), Page 13:
"The rolling of dice is a convenient way to represent unknown variables or to assist the referee in making decisions. Feel free to modify the results if you do not like the way they turned out."

"If you are in a fair fight, you are doing something wrong." Old military maxim. As GM, it's good to drop hints at what the likely outcomes are going to be. Even then, use some napoleonic tactics, or the antagonists have a light mortar? Can be extremely deadly, so I find myself fudging even that. Then again I have seen GM's actively fudge things against players who are doing things too well.

#### clearstream

##### (He, Him)
45% is certainly no more helpful than "9 out of 20".

But I'm not talking about that, I'm contrasting transparent to opaque odds.

Something like "45%" or "get +1 to your d20 roll" is transparent.

Something like "get +1 to your 2d6 roll" is immediately less transparent, since it requires an understanding of bell curves. Same with D&D's advantage "roll two dice, pick the best" (which gives the most benefit when your original success rate was 50%).

And then there's the profoundly opaque resolution systems of dice pool games: How likely are you to "roll five dice, get two successes"? Which is best - "gain an extra die" or "get +1 to all your dice"?

Cheers
True enough. What do you think the benefit of those mechanics are? Why do people use them?

#### Blue

##### Ravenous Bugblatter Beast of Traal
I meant that as a "solvable" monster, simply avoiding it isn't solving it, which is why you don't get xp (or treasure or magic) from avoiding said monster. Thus the monster isn't "solvable," which was your original assertion. Some things simply cannot be solved.

That's why I moved away from monster XP. It's full of actively harmful behavior modification, both in what players are rewarded to do do and what they aren't.

If the goal is to get into the kingpin's chamber and deal with him, killing guards, bribing guards, sneaking past guards, taking a trap-filled secret route that avoid the guards, tricking guards - all of those deal with challenge of "getting into the chamber". "Killing guards" is just one equal way among many of accomplishing it and shouldn't carry more or less rewards than others. Solving for "monster" instead of solving for plot leads in the wrong direction to me.

#### Blue

##### Ravenous Bugblatter Beast of Traal
I'll give another example of risk/reward calculations.

In Star Trek Adventures, you are typically rolling 2d20 for most checks. Sometimes 3 (or possibly more) and sometimes 1d20, but, typically, you roll 2d20. Now, in the rules, if you roll a 20 on a check, you get a complication. But, and here's the kicker, the "complication" is largely left up to the DM to define.

So, think about it for a second. On 2d20, you have a 9.75% chance (just a smidge less than 1 in 10) of rolling a 20. If you have a group of 5 players, it's pretty much guaranteed that SOMEONE is going to roll a 20 every round.

This is not picking on you, but just a great example.

Even people who understand probability (shown from Hussar's chance at a complication for a single roll) to misjudge it. The chance of it happening in a round with 5 rolls is 1 - ( .9025^5 ), showing that only ~40% of the rounds have complication(s).

And for math that is harder: 5 tries 9.75% success - Wolfram|Alpha

Multiple tests is an easy place for even savvy people to lose track of the actual probabilities.

#### CapnZapp

##### Legend
True enough. What do you think the benefit of those mechanics are? Why do people use them?
Assuming you mean the latter, opaque, ones, I have no better answer than the one I already gave:

As far as I understand it, lewpuls, the opaque odds of dice pool games (e.g. Vampire) was intended to be a feature not a bug. That is, people play that game because they don't want to know the (exact) odds.

But I guess (and this is not intended to be a personal insult to anyone) if you don't do math, playing a game with opaque probabilities (figuring out the odds in dice pool games is fiendishly difficult) evens out the odds compared to a friend that do math...

In short - I have no idea.

#### Hussar

##### Legend
This is not picking on you, but just a great example.

Even people who understand probability (shown from Hussar's chance at a complication for a single roll) to misjudge it. The chance of it happening in a round with 5 rolls is 1 - ( .9025^5 ), showing that only ~40% of the rounds have complication(s).

And for math that is harder: 5 tries 9.75% success - Wolfram|Alpha

Multiple tests is an easy place for even savvy people to lose track of the actual probabilities.

Heh. This really does illustrate the point nicely doesn't it. Of course 5 tries is about 40% chance. I was thinking that it was 9.75/die, not /try.

Which really does roll it back around to the point I made about DM's making judgements at the table. It's very, very easy to get it wrong. Particularly when you're doing a snap judgement at the table. After all, a 60% chance of success isn't hard, is it? Hard should be down around 30% chance of success. But, the thing is, 60% success rates are pretty difficult. If you are failing to do something about 1 try in 3, that's a pretty hard task that you're attempting.

But, most DM's don't grok that because it's not really very intuitive. And, then, you start adding in egos at the table, and it can get messy, REALLY quickly.

#### clearstream

##### (He, Him)
Assuming you mean the latter, opaque, ones, I have no better answer than the one I already gave:
I want to understand the motives for a thing in order to critique that thing. Thinking about dice pools, I can appreciate their tactile qualities - it's really pleasant to pick up a bunch of dice and roll them - and the way they push me toward an intuitive and approximate rather than analytical and exact idea of the odds. A long time ago, and ultimately unsuccessfully, I designed an RPG in which the dice represented spirits that players can use and exhaust; each having faces with unique numberings and additional game effects. There have of course been many other forays into this design space.

It seems like the purpose of dice pools might be exactly contrary to the assumptions of the OP, whose analysis appears built upon a notion that players should, and need to, know their odds. The question it makes me ask is, why? Why does it really matter that players know their odds? I could come back with a notion that they should not know their odds, but only when their chances are improved or worsened, and when they are stronger or weaker. I could even think that knowing the odds in any exact sense is a chimera.

Not that I am not defending any specific position here. I'm digging into when and why one might not want to know the odds. The OP offers a one-dimensional analysis of the subject, notwithstanding that I appreciate the thought and effort in writing, and that it has prompted an interesting discussion.

#### clearstream

##### (He, Him)
Heh. This really does illustrate the point nicely doesn't it. Of course 5 tries is about 40% chance. I was thinking that it was 9.75/die, not /try.

Which really does roll it back around to the point I made about DM's making judgements at the table. It's very, very easy to get it wrong. Particularly when you're doing a snap judgement at the table. After all, a 60% chance of success isn't hard, is it? Hard should be down around 30% chance of success. But, the thing is, 60% success rates are pretty difficult. If you are failing to do something about 1 try in 3, that's a pretty hard task that you're attempting.
I feel like an important DM skill is sizing how much you are increasing or worsening the odds. Say you concede to a group check instead of individual checks: how much difference is that making? Suppose you apply disadvantage or even a rare flat minus or penalty die, is that going to be a big or small change? Sizing doesn't need to be exact to be useful.

When you say that hard is 30% chance of success, you are saying a number of things. On the surface, you are saying that on roughly three times that I try this, I only succeed once. If using a d20 and the consequences of failure rules from the DMG, you are also saying that about half the time something very bad happens. You're saying that bardic inspiration will increase your chance of success to close to a coin-flip. At 60%, bardic inspiration nearly halves the count of possible worlds in which you fail. And then there are contests, where the bar is not necessarily fixed.

The difference between success on 15-20 and success on 9-20 is a complex beast. What's at stake also matters. A 40% chance of perma-death versus a 60% chance of losing 1gp say. That context is crucial.

But, most DM's don't grok that because it's not really very intuitive.
Perhaps in the end all that needs to be grokked is exactly that which is intuitive. This is where I see some virtue in the OP. A DM needs a sense for when chances are good or bad remembering that they compound over trials, and against that what is going to produce a big change, and what a small change, so that they can consciously apply whichever they want.

#### Hussar

##### Legend
/snip

It seems like the purpose of dice pools might be exactly contrary to the assumptions of the OP, whose analysis appears built upon a notion that players should, and need to, know their odds. The question it makes me ask is, why? Why does it really matter that players know their odds? I could come back with a notion that they should not know their odds, but only when their chances are improved or worsened, and when they are stronger or weaker. I could even think that knowing the odds in any exact sense is a chimera.

Not that I am not defending any specific position here. I'm digging into when and why one might not want to know the odds. The OP offers a one-dimensional analysis of the subject, notwithstanding that I appreciate the thought and effort in writing, and that it has prompted an interesting discussion.

When you say, "players" are you including the GM/DM in there? Because I'd argue that it's very important that the person running the game is cognizant of the odds of success. Primarily because when the person running the game doesn't know the odds, then most of the time, the odds get stacked up against the PC's, because the DM/GM want's to "challenge" the characters, typically. The trick is, when the GM doesn't understand the odds, the line can very quickly shift from "challenging" to "virtually impossible."

Like I said earlier, the whole "I scout ahead" thing typically goes tits up because the GM doesn't understand that when he forces repeated checks, and any failure is a catastrophic failure, we've moved from difficult to impossible.

#### Schmoe

When you say, "players" are you including the GM/DM in there? Because I'd argue that it's very important that the person running the game is cognizant of the odds of success. Primarily because when the person running the game doesn't know the odds, then most of the time, the odds get stacked up against the PC's, because the DM/GM want's to "challenge" the characters, typically. The trick is, when the GM doesn't understand the odds, the line can very quickly shift from "challenging" to "virtually impossible."

Like I said earlier, the whole "I scout ahead" thing typically goes tits up because the GM doesn't understand that when he forces repeated checks, and any failure is a catastrophic failure, we've moved from difficult to impossible.

I realize I'm kind of changing the subject slightly here, but I think stealthing around is actually one situation where multiple checks can often be appropriate. The key each success needs to yield something beneficial - whether it's a good ambush position, access to something that wasn't previously available, or previously hidden information. It's perfectly normal, for example, to make a roll to sneak closer to a booth where a conversation is overheard and the rogue can see an alarm button, and then force another roll if the rogue wants to try to make it over to the alarm and disable it.

#### clearstream

##### (He, Him)
I realize I'm kind of changing the subject slightly here, but I think stealthing around is actually one situation where multiple checks can often be appropriate. The key each success needs to yield something beneficial - whether it's a good ambush position, access to something that wasn't previously available, or previously hidden information. It's perfectly normal, for example, to make a roll to sneak closer to a booth where a conversation is overheard and the rogue can see an alarm button, and then force another roll if the rogue wants to try to make it over to the alarm and disable it.
I think by RAI (and maybe RAW) the rogue makes one check and that rides until stealth is broken. So in the case outlined, there is mechanically no requirement for a second check unless stealth was broken while eavesdropping. Possibly with the sort of concerns @Hussar has raised in mind.

#### CodeFlayer

##### Explorer
I think the math matters if the table decides it does. This is because D&D can be played in many, not necessarily disjoint ways, such as (wargame, story, TotM, ...).

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