What Are the Odds?

Probably the central conceit of the dice part of the rules in Four Colors al Fresco is that it’s the step size between the dice, not the actual sizes of the dice, that matters. But is this true?

Mostly.

Four Colors al Fresco isn’t a number-cruncher’s dream system, but I still want the rules to actually do what they supposedly do and thus stay out of the way. I’ve played some “story-oriented” RPGs (and even some that were actually concerned with the math) which didn’t stand up to scrutiny, so I don’t expect you to just take my word for it. So here is where I show my work. 

All the rulebook says–and is going to say–is that the probabilities work. This really isn’t the sort of game for gamers who are particularly concerned with the math or balancing point buys. I figure that those who want to know the probabilities don’t want this game, and those who want this game don’t particularly care about the probabilities. And for the few who are interested in both, I don’t want to set the wrong tone for Four Colors al Fresco. 

However, it’s a question that comes up once in a while, so I thought I’d put the answer here for those who want it.

The short answer is that we figured the probabilities back in ’00 before settling on the dice step sizes, in order to make sure they are all roughly even. 

The Long Answer

The important caveat is that this talks about the dice sizes. This is to support the conceit that a d4-d8 step is equivalent to d8-d12 or, more broadly, that two characters who have, say, d6/d8/d10/d12 and d10/d12/d16/d20 are roughly equivalent when rolling the dice–that you can choose whichever option and not significantly disadvantage your character. Of course, as the rulebook says, they’re not exactly the same. Most importantly, the character with smaller dice will have an advantage when dealing with Hindrance dice, while the character with larger dice will have an advantage when dealing with Bonus dice. For this reason, the rules suggest that, if you don’t have any other reason, err on the side of smaller dice. 

The other significant caveat is that this is only looking at how 2 dice compare–once you roll 4 or more dice, it gets really complicated. The basics obviously still hold, but simply comparing each pair of dice is a little like solving a multi-body gravitational problem one pair at a time. 

So, on to the math. 

Definitions

First, some labels. N will be the larger die of a pair of dice and n will be the smaller one. N > n will be referred to as the expected result. N < n will be the unexpected result, while N ≤ n (or {N < n} + {N = n}) will be the inverse result. 

Now, the probabilities. If N is the larger die and n is the smaller die, then

  • the odds of the smaller die having the smaller result are (2N-n-1)/2N,
  • the odds of the larger die having the smaller result are (n-1)/2N,
  • and the odds of a tie are 1/N.

It’s interesting to me that the odds of a tie are independent of the smaller die. 

The possible die scores in Four Colors al Fresco are d3, d4, d6, d8, d10, d12, d16, d20, d24, d30, in that order. “1 step” refers to any 2 dice that are adjacent on that list (like d8 & d10), “2 steps” would be two dice separated by another on the list (such as d10 & d16), and so on. Oh, and small rolls are better. 

Now on to the results. Plugging the numbers in, for the most part all the equivalent steps are in fact the same, though it depends on how you measure it: Is the question “what are the odds of a smaller die giving a smaller result?” or “what are the odds of a smaller die giving a larger result?” These aren’t equivalent (or inverse, rather), because you can also have a tie and the odds of a tie go up the smaller the dice involved. So while the likelihood of the expected result (the smaller die producing the smaller result) is remarkably consistent, the likelihood of the unexpected result (the smaller die producing the larger result) decreases noticeably as the dice involved get smaller. 

1 die step odds

But even then, the odds of a tie don’t really jump until you involve a d4, and therefore the odds of the unexpected result stay pretty steady until you get to a d4 or smaller. As you can see, two dice that are one step apart have very similar odds of the expected result–mostly around 55%, with d3-d4 being the outlier at 50%. Due to the tie effect, the odds of the opposite result varies a bit more, from 40% to 25%.

2 die step odds

Looking at dice two steps apart, the odds are similarly consistent, with the odds all right around 2/3rds for the expected result (the actual odds range from 63% to 69%–or ±3 points). Again, the odds that the larger die gives the smaller result are still very similar, but not quite as consistent–this time they range from 32% to 17%. And, as expected, all of these results are more skewed than when the dice were closer in size, which is also how we want the rules to work. 

Going on to bigger steps continues the trend–which makes sense, since all of the single steps are so similar, and that’s just compounding them. The odds of the expected result are bigger for bigger steps, but two pairs of dice of different sizes will have roughly the same odds of the expected result if each pair is the same number of steps apart. 

Given the overall loose nature of the rules, I’m comfortable saying that the actual behavior of the dice is close enough that it fulfills the intended behavior. It’s not perfect, but this also isn’t ShadowRun or D&D–or even Vampire: the Masquerade–and therefore the expectation isn’t probabilistic precision, but just a lack of overtly anomalous results. 

And, back to our questions, another way of looking at these odds would be “what are the odds of a smaller die not giving a smaller result?” This is actually the inverse of the expected result, and therefore is remarkably steady across die sizes, provided the step size is the same. This is lumping together the tie and unexpected results, and the reason this is a sensible thing to ask is because of how the dice behave in play. In play, you’re hoping your Favored die comes up as the smallest and your Opposed die comes up as the largest. So, just looking at those two dice, a tie is almost as bad as the inverted order (large die smaller than small die).

And, more generally, it means that you can roughly generalize just from knowing the die sizes, and the odds are that you’ll get the results you expect. The only part that might be a surprise to a player is that dice one step apart are pretty unpredictable–55/45 isn’t exactly reliable odds. 

So, to sum up:

  • Small die beats large die the majority of the time.
  • If they’re very close together (only 1 step apart) the small die has a very small edge–with d3/d4, it’s actually a toss-up. 
  • The further apart the dice are, the bigger the edge becomes.
  • The odds of a tie are independent of this step gap, are dependent only on the larger die, and increase the smaller it is.
  • Two dice the same number of steps apart have roughly the same odds, regardless of what specific die sizes we’re looking at. There’s a little variance, but it’s really only when a d4 or d3 is involved that that variance is enough to even notice, and I’m comfortable saying that it’s still not enough to matter. 

 

 

 

 

 

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s