“A 12 or less? That’s good… right?”

If you come from one of those popular games that uses a twenty-sided die to whack monsters and roll other checks, and you step up to one of several games that uses three six-sided dice instead… you might feel a little lost.

We deal with percentage-based probabilities every day: “30% chance of rain” and all that. So given the obvious fact that each number on that twenty-sided die has a 5% chance of occurring, it’s easy to mentally convert “Roll 19 or higher to succeed” into “10% chance of success”, or to understand “You shrug off the poison on a 12 or higher” as “11 or lower fails… so 55% chance of dying”. Simple.

However… “Roll 9 or less” on three six-siders? For newcomers to games like GURPS, that’s not so intuitive. But maybe we can make it so.

Your chances on 3d6 are discussed abundantly online, and even within properly-done rulebooks themselves. Check out p. 171 of GURPS Basic Set for an example.* Give newcomer players easy access to a table like that, and they’ll weigh the odds like a Vegas pit boss before trying to leap that lava pit.

* Nitpick: GURPS places those percentages in the “Skills” section of the book, and describes them as the percentage chance of success on a skill roll. But the numbers apply to any “roll equal to or under on 3d” roll, not just skills.

I’m offering nothing new by repeating those numbers below. But I’ll expand the table of probabilities into something you may find more handy once in a while, and I’ll add something players may like: rules of thumb for gauging the effects of modifiers, no table needed. (While I know I’m primarily talking to experienced GURPS GMs, I’ll try to write it up in a way new players can appreciate, too.)

Important note: All of this page addresses pure die rolls alone, with no consideration of system-specific additions like “critical” meanings for certain rolls. GURPS, for example, has critical success/failure rules that change the effect of high and low rolls, but the relevant targets vary with circumstances (such as skill level). Some GMs further tweak GURPS critical rolls with house rules; others (Hi!) really aren’t fond of the critical roll rules to begin with.

For simplicity, I’m ignoring the complication of “special” rolls. For your games, be sure to explain to your players (or ask your GM!) how such rolls affect things.

“Bo-o-ring. Do I have to read a bunch on this?”

Not at all! Feel free to jump ahead and just grab a summary. Take your choice:

• The whole schmear: Probability tables and all that. Good for nerds. No jump; just keep reading.
• Eyeballed analysis: A distillation for GMs (or dedicated players) who don’t need all the details, just a fair picture of what mods mean for success and failure. Click here.
• Simpler guidelines for players: An even more boiled-down summary that any player should find handy. Click here.
• Simplest overview for players: A mere few sentences, to arm any new player with a sense of what to expect from 3d6 rolls. Click here. 

What are the chances?

My little table below shows the sum of 3d6 on the left. That’s your target in a game like GURPS: you want to roll that number or lower to succeed at a task.

The five columns of percentages represent your probability of rolling what’s indicated in the respective column labels. They’re detailed below the table.

What it means to roll…

the target or lower: This is your probability of success – that is, the probability you care about most! These are the same percentages shown in GURPS Basic Set. (In GURPS terms, you could also call this “success by 0 or more” – which is simply “success”.)

If your game uses 3d6 differently from GURPS, one of the other columns may represent your chance of success (or you may need to make your own table).

higher than the target: This is the probability that you will roll higher than the target number and fail the task. The game’s rules and players may not focus on this percentage very often, but it can be interesting to examine. (In GURPS terms, you could also call this “failure by 1 or more” – which is just “failure”.)

the target exactly: I’ve toned down the appearance of this column and the next two, as what they represent isn’t often given attention in the game. But should you want to know the probability of rolling that 14 exactly, here you go. (In GURPS terms, this is your probability of “success by exactly 0”, which some rules will reference.)

lower than the target: This is not a probability you’ll often be concerned with; it’s here for completeness. (In GURPS terms, this is your probability of “success by 1 or more”.)

the target or higher: Again, this is for completeness. (In GURPS terms, this is your probability of failure plus your probability of “success by 0”, should you want to know that.)

Tot ’em up

The following observations score much higher on the obvious scale than on the interesting scale. But as long as we’re here:

• The probabilities of “the number or lower” and “higher than the number” add up to 100%, as they must. The same goes for the probabilities of “lower than the number” and “the number or higher”.
• Just as obviously, adding the probabilities of “the number exactly” and “lower than the number” gets you the probability of “the number or lower”, and adding the probabilities of “the number exactly” and “higher than the number” yields the probability of “the number or higher”.
• Newbie tip: The average roll on 3d6 is 10.5, smack between 10 and 11. Statistically, half your rolls will come up 10 or under and half will come up 11 or higher. (Hence the 50/50 chance of success when your target is a roll of 10 or under.)

Okay. Now we move on to interesting:

“Just a +1 to hit? Does that even matter?”

Why, yes, it does. You might be surprised by how much a small bonus or penalty matters.

Modifiers and success

Take some targets and check your chances of success on the above table. For example, a target of 6 gives you a 9.25% chance of success, a target of 10 a 50% chance, and a target of 14 a 90.74% chance.

Now add some modest bonuses or penalties to see how things change.

• For that target of 6, a wee -1 penalty means you now have to roll 5 or less, meaning your chance of success drops from 9.25% to 4.62% (a drop of 4.63 percentage points).
• For that target of 10, a bonus of, say, +2 means you now roll 12 or less to succeed, boosting your chance from 50% to 74.07% (an increase of 24.07 percentage points). That’s a much bigger boost in percentage points than you’d get with a +2 bonus on a d20 roll!
• For that target of 14… hmm, let’s try a substantial penalty of -3. You now have to roll 11 or less, which cuts your chance of success from 90.74% to 62.5% (a drop of 28.24 percentage points).

That’s interesting to see, but looking at percentage points alone doesn’t tell the whole story. It’s at least as interesting to look at the size of that change relative to the chance of success itself – i.e., at how the modifier multiplies your chance of success. Just take your percentage chance of success with the modifier, and divide it by your percentage chance of success without the modifier.

• For that target of 6, the -1 penalty cuts your chance of success by less than 5 percentage points, which doesn’t sound bad – except for the obvious fact that this halves your chance of success (4.62% / 9.25% = 0.50). If you attempt this feat multiple times, the “wee” -1 penalty takes your already-few successes and cuts them in half. Harsh!
• For that target of 10, the +2 bonus yields a multiplier of 74.07% / 50% = 1.48, meaning this bonus boosts your chance of success by roughly a half. Perform the feat, say, 20 times, and where you would have expected about 10 successes without the bonus, you can now expect about 15 successes. Sweet!
• And then there’s that target of 14. The -3 penalty yields a multiple of 62.5% / 90.74% = 0.69, cutting your chance of success to a little better than two-thirds of what it could have been. Maybe that’s not as bad as you’d expect from a penalty that size, but it still smarts!

Modifiers and failure

Let’s look at that last example from a different angle. Whether you focus on the -3 penalty’s effect as a multiplier or as a loss of percentage points, it’s clearly an unwelcome penalty. Yet it leaves you with a better than even chance of success, so it’s not that bad, right?

No argument here. But just for fun, take a look at your chance of failure (the figures in red). Without the penalty, your chance of failure is 9.25%. With the penalty, it’s 37.5% – four times as high! If you were to perform this task, say, 20 times, the penalty means you’ll go from failing perhaps 2 times to failing about 8 times. Quadruple the failures! Ouch!

Multipliers ahoy!

Just for kicks, here are multipliers worked out for a range of bonuses and penalties.

Note: If you’re a player or even a GM, you do not have to play with these figures or even have the slightest interest in them! I post them here for those looking to divine the simple secrets of 3d6 rolls – and so I can derive the simple rules of thumb that follow. Feel free to skip right on past these tables!

First, here’s how bonuses multiply your chances of success (in black) and failure (in red):

Next, here’s how penalties multiply your chances of success and failure:

There you go. Please don’t mess with this silliness during play (if ever). If you only want to glean what’s interesting, give a scan and look for the big game-changers: the numbers far above 1.0 or close to 0. (Examples: Note how, at a target of 7, a +3 bonus multiplies your chance of success better than three-fold. At a target of 11, a -4 penalty slashes your chance of success to about a quarter.)

Eyeballed analysis for GM types

We can simplify the above into rules of thumb to help a GM (or odds-loving player) eyeball the effect of modifiers. Perhaps something like this, with plenty of rounding and other generalizing:

If your target is 3 to 5:

• Any bonus is awesome, multiplying your chance of success many times over. Even a +1 bonus at least doubles your chance!
• Any penalty is devastating, either eliminating your chance of success or (at best) reducing it to nearly nil.

If your target is 6 to 7:

• Any bonus is great. Even a +1 bonus multiples your chance of success by better than 1.5; a +2 bonus better than doubles it.
• Any penalty is harsh at best. A -1 penalty roughly halves your chance of success, a -2 penalty roughly quarters it, and a -3 penalty or worse pretty much eliminates it.

If your target is 8 to 11:

• Any bonus is nice. A +1 bonus multiples your chance of success by less than 1.5 (but cuts your chance of failure to about three-quarters). A +2 bonus falls short of doubling your chance of success (although it roughly halves your chance of failure).
• Any penalty is painful at best. If your target is 10, for example, a -1 penalty cuts your chance of success – only 50% to begin with – to three-quarters; a -2 penalty, to half; a -3 penalty, to a third; a -4 penalty, to a fifth; and a -5 penalty, to a tenth.

If your target is 12 to 15:

• Any bonus is welcome, if mostly unnecessary; large bonuses become overkill given your already-high chance of success (unless degree of success matters, as in many GURPS rules). However, each +1 bonus significantly slashes your increasingly small chance of failure.
• A small penalty is unwelcome and a larger one will painfully drop your chance of success to iffy levels. Each -1 penalty significantly multiplies your initially small chance of failure.

If your target is 16 or higher:

• Any bonus is superfluous (unless degree of success matters).
• A small penalty is nearly negligible, though a larger one will see your chance of success plummet from nearly-certain to merely comfortable. At the same time, each -1 penalty greatly multiplies your initially tiny chance of failure.

Simpler guidelines for players

Let’s shorten the above even more, for players who just want a rough understanding of key targets and small modifiers in a “roll under” 3d6 game. (For targets in between the below, extrapolate. For the effects of larger modifiers, think “like the effects of +/-2, but more so!”)

• If your target is 6 or lower, your initial chance of success is very poor (9% or much lower). Bonuses greatly multiply this (tiny) chance of success, while penalties quickly eliminate it.
• If your target is 8, your initial chance of success is poor (25%). A +1 bonus multiplies it by 1.5 and a +2 bonus doubles it, while a -1 penalty cuts it to two-thirds and a -2 penalty cuts it to one-third.
• If your target is 10, your initial chance of success is iffy (50%). A +1 bonus multiplies it by 1.25 and a +2 bonus multiplies it by 1.5, while a -1 penalty cuts it to three-quarters and a -2 penalty halves it.
• If your target is 12, your initial chance of success is high (75%). Bonuses begin to have fairly modest effect on your chance of success, but a +1 bonus cuts your already-low chance of failure to two-thirds, while a +2 bonus cuts it to one-third. A -1 penalty, meanwhile, cuts your chance of success to about four-fifths, while a -2 penalty cuts it to about two-thirds.
• If your target is 14 or higher, your initial chance of success is very high (90% and up). Bonuses have very modest effect on your chance of success, but a +1 bonus cuts your already-very-low chance of failure to two-thirds, while a +2 bonus cuts it to one-third. A -1 penalty, meanwhile, cuts your chance of success to about nine-tenths, while a -2 penalty cuts it to about four-fifths.

Simplest overview for new players

Final summarization: Forget any numbers. Just make sure every newcomer understands this:

• When your roll-under target is low (i.e., success is unlikely), even small bonuses and penalties have a huge effect on your low chance of success.
• When your roll-under target is high (i.e., success is already likely), bonuses and penalties have less drastic effect on your chance of success (although they greatly affect your remaining small chance of failure).
• No matter what you’re rolling against, even small bonuses are always welcome (as are even small penalties, when they’re penalties on the other guy). These small modifiers are always important, too, when degree of success matters!

Now go roll dice!

The big wrap

The modifiers a GM hands out may look trivial, but don’t underestimate the big power of a humble +1 or -1 – the effect will often be more pronounced with a 3d roll than with a d20 roll. Accept every little bonus you can, stick your foes with every little penalty you can, and make those dice work in your favor!

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Header image: A tonsured monk invites a scruffy traveler to roll the bones?

Not sure about this character’s clerical bona fides; he looks like a hustler. (The Ray-Bans aren’t helping any.)

14th century, German or Swiss (according to the fun weird medieval guys account).