“…the dice are your tools. Learn to use them properly, and they will serve you well.”
Gary Gygax, 1st Ed. Dungeon Master’s Guide, 1979
The last post dealt with the difference between the frequency and probability of items on a random encounter table, and introduced a simple bell-curve table using 1d4+1d6. We’ve gotten far afield from creating and populating a mini-setting, but the discussion about tables and die rolls is worth having, so I’d like to spend one last post talking about some other (ahem) facets of the dice that may help you as a GM.
Dice Basics
Some of this is no doubt familiar, but I’m hoping that even the most experienced among you can grab a few bits of value.
When rolling a single die, the number of possible results [R] equals the number of sides [S]. The probability of rolling any single result (as a percentage chance) equals [1 / S]. Thus, when rolling 1d6, there is a 16.7% chance of getting a 1, 2, 3, 4, 5, or 6.
When rolling multiple dice, the number of possible results [R] equals the highest possible value [H] minus the lowest [L], plus 1 [(H – L) + 1]. Thus, when rolling 2d6, there are 11 possible results (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12).
The mean value of any die roll (regardless of how many dice you’re rolling) is the highest value plus the lowest, divided by 2 [(H + L) / 2]. Thus, the mean of a 2d6 roll is 7; the mean of 1d4+1d6 is 6; the mean of 5d4 is 12.5, etc.
Bell Curves
When rolling multiple dice, you automatically skew the results along a bell curve, where the values around the mean have a higher chance of occurring than values at the high and low extremes.
This is because when you add the result of each die, there are more combinations that sum to median values. Conversely, there are fewer die combinations that add to values at the high and low ends. This is well-known to most gamers, who are used to rolling 3d6 for ability scores. When you roll 3d6, for example, there is only dice combination that gives a 3 {1,1,1} or an 18 {6,6,6}, but there are 27 dice combinations that add up to a score of 10.
The total number of dice combinations for any roll is the product of the highest value of each die. Thus, 1d6 provides 6 combinations; 2d6 provides 36; 3d6 provides 216 combinations, 1d4+1d6 provides 24 combinations, etc.
The probability of rolling a particular value equals the number of combinations that add up to that value divided by the total number of combinations possible. Thus, when rolling a 3d6, the chance of rolling a 3 is [1 / 216] or 0.5%; the chance of rolling a 10 is [27 / 216] or 12.5%. Chronicler Isiah over at The Dark Fortress has a great set of 3d6 probability charts illustrating this familiar curve.
Calculating the Odds
You may have seen tables that lay out the value and probability of die results. The problem is figuring out how many dice combinations add up to each value—it’s time consuming, can be error prone, and all sorts of dull.1Or, you can ignore all of this nonsense and get on over to AnyDice.com, which will spit out the probability table of any die combination you want. Thanks again to Roger for bringing this to my (our) attention.
Good news is that it’s very easy to create your own probability tables for rolling a pair of dice, which I daresay is sufficient for most encounter tables. I say this for one simple reason: the more dice you roll, the more results your table will produce, which means the more “slots” you have to populate.
Back in the day, TSR provided an encounter table template using 1d8+1d12, granting a range of 2-20. It has a nice curve, but it means you have to supply 19 results. If you’re using nested encounter tables, that can be too beaucoup. Better (says I) to stick with smaller ranges, like the 1d4+1d6 combo I suggested earlier (wherein only 9 results are required, which is just faster and easier to work with).
But the topic here is figuring out the probabilities associated with the dice combination you want. Note that this only works for 2d-something. If you’re rolling 3d6, 4d5, or 10d8+9d3, the guidelines below won’t work and you’ll have to do more math. And I think we all know how I feel about that…
Let’s start with a basic 2d6 toss. We know that there are 36 possible roll combinations (e.g. {1,1} {1,2} {1,3}, etc.). Here’s a quick way to figure out the curve.
Start by creating a table, with columns for Values, Combinations, and Probability. The values for 2d6 look like this:
Determining combinations is easier than you might think. The secret is this: at the roll’s mean value, the number of combinations equals the highest value on the lowest die rolled. For 2d6, the mean value is 7, and there are 6 combinations that get you there. This is the high point of the curve, and the number of combinations gets lower on either side, until you get to 1 combination on each of the highest and lowest values.
In the Combinations column, enter 1 for the first row (i.e., there is 1 combination that grants a value of 2). Enter 2 in the second, 3 in the third, etc, until until you reach the median value. Then, reverse it, so that your table looks like this:
Finally, determine probability by dividing the combination figure by the total number of combinations. For 2d6, the total combinations possible is 36, so do some math and fill in your table like so:
But what if you’re doing something funky like 1d8+1d12? It’s the same procedure, with a slight twist:
- There are 96 roll combinations (8 * 12 = 96)
- The result range is 2-20
- The median value is 11 [(2 + 20 = 22) / 2 = 11]
- There are 8 combinations that garner the median value (8 equals the highest value on the lowest die rolled)
Create your table as above, filling in results from 2-20. Enter your combinations the same way, starting with 1 and progressing to 8. However, you’ll note the progression doesn’t go all they way to 11—it stops at 9, like so:
Now what?
Here’s what: Start filling in the combinations at the other end of the result set, like this:
What to do with results 10, 11, and 12? Well, not surprisingly, they also have 8 combinations each. It’s a quirky bit of math that occurs because you’re using different die types. As a result, the curve peaks at several points, but still along the median value, which in this case is 11.
Fill in those bits, and your final table looks like this (I’ve included the actual combinations as proof of the system):
Final Words
OK, some stupid dice tricks that I hope are useful to you. Remember that the bell curve probability method described above only works for 2d-something combos.
I think this wraps up all that I want to say about die rolls and probabilities. Next post, I promise we’ll get back to actual encounter tables.
Another stupid dice trick: you can get 1d7 by rolling 1d8 and dropping and rerolling any 8’s. They’re mathematically equivalent. Other odd values work this way too.
Playing DCC and can’t find your d14? Roll 1d4/2 x 1d8 [drop 8s]. The first chapter of the 1e DMG has all sorts of spiffy tricks to play with dice.
@Roger3 Cool. Here’s one for 1d3: on a proper d6, opposite sides add up to 7, so if you roll a 1d6, results of 1, 2, or 3 are read normally. A roll of 4 = 3, a roll of 5 = 2, and a roll of 6 = 1. Just to save you the d6/2 and rounding bit (This trick from Sir Larkins at RPG Corner: http://shirosrpg.blogspot.com/2012/01/how-im-rolling-my-three-sided-dice-from.html).
On your d7 trick, does the same math apply for rolling a 1d9 with a d10 (i.e., roll 1d10 and drop any 10s)?
@Erin D. Smale
Yeah, pretty much any dice can be ‘shortened’ by dropping results from the highest to the lowest.
Yup. Basically, you’re trading potential extra die rolls for lower numbers. 1dX drop all X’s is always going to be mathematically equivalent to 1dX-1
I’ve often thought about a game mechanic that lets players specify their own level of luck. The base roll would be 2d8 + modifiers. But you can swap dice to 2d4+4, 2d6+2, 2d10-2, 2d12-4. The median roll value would be the same, but the high and low value would be variable. Using the lower dice would mean playing it safe. You don’t have as much of a chance for a big success, but you don’t risk as big a failure. Using big dice is swinging for the fences – you can generate a big total, but you can easily generate a very low total. Of course this means the game would need to reward degree of success and penalize degree of failure. For example, succeeding by more would inflict more damage and failing by more would give you a penalty against the return attack.
@Philo Pharynx : I like this idea – giving players more control of the dice, as risk vs. reward, is great. In Chimera, I could see this as a Clutch Situation thing: instead of spending 1 CS for an extra die, maybe each CS increases the die you roll by 1 step (i.e., 1d4 -> 1d6; 1d6 -> 1d8; etc.). Or, more appropriately, the player gets to choose which option when he rolls.
Any more of this dice talk and I may have to convert my Hedral parody races to Chimera. Behold, the Tetras.
http://www.lejendarylands.org/Forums/viewtopic.php?f=6&t=4&p=1895
@deimos3428 Sorta like modrons, but not.
BTW, good to see you back!