APPENDIX: NUMBER CRUNCHING

Beware all who enter - this section was written (and will probably be edited by) a maths teacher. Take from that what you will.

One of the (many) fantastic things about Elfball is that the probabilities are more complex than some other games, many of which involve basic d6 counting. Even without the complexity of conditional successes, the fact that one of the faces is basically a negative success makes life more interesting. In a challenge you can roll up to 6 dice, which means there are 6 x 6 x 6 x 6 x 6 x 6 possible outcomes, which comes out as 46,656 different combinations. Now some of these are the same, in that there are two bull's-eyes and two blanks, but this doesn't really make life any easier.

The Basic Data

The easiest way to start the analysis is with a spreadsheet listing all the outcomes, and the total successes that come from each combination. If you do that you get the first table:

Successtable.jpg
Raw Figures

Notice that the total at the bottom is the total number of flops, so the sum of all the negative success outcomes. Dividing all of these by 46,656 gives the percentage probability of each of the outcomes, given in the table below. So for example, if you roll 4 dice, the probability of getting exactly 3 successes is 17.9%, shown by the highlight. Just to be sure, this means that if you rolled 4 dice 100 times, around 18 of those times (or just under 1 in 5 times) you would get 3 successes.
Successtablepercenthighlight.jpg

Now, getting exactly the right number of successes doesn't actually matter, what usually matters is whether you can get at least a given number of successes, without flopping, which is where the (nicely formatted) table generated by GrumpyGrizzly and Antipixi comes from, as shown below. The flop percentages are those shown above, as are the zero successes. The rest of the numbers are the probability of getting at least that number, so for example the 2 success number on 2 dice is the sum of 2, 3 and 4 successes.

table2.jpg

Example 1 - Interpreting the Table

While this is obviously lots of fun, what is it really useful for? Well, it firstly tells you whether or not to get cross when something goes wrong. For example, if you need to roll a 6 to go first in a game, and fail, or need to roll snake eyes to win a craps game, you know that the probabilites are about 1 in 6 and 1 in 36 respectively. In Elfball, if your Widowmaker is tackling a Striker without the ball, they will be rolling 5 dice and need 5 successes to damage that striker (2 more than (the dodge characteristic-2) + 1). The probability of getting this is 8.1%, which is about 1 in 12, so similar to the probability of rolling an 11 or 12 on 2 dice (ie not very likely).

Example 2 - Dash Strategy (Part 1)

Dashes are fairly common in Elfball, so understanding the probability involved can make a marginal difference to your chances of winning. You have a certain amount of momentum to spend, but how do you spend it? You can compare two main options - roll all the dice, or roll all the dice except one, saving that one to replace in case you get zero successes. The probabilities come out as shown below:

dashstrategy.jpg

The analysis I've done is straightforward. The dash challenge success figure is the raw figure from the table above. The row below says that, for example, instead of rolling 2 dice, you roll 1 dice. If you get 1 or 2 successes on that dice (50% probability), great, you've passed the challenge. If you get zero successes however, you have the chance of replacing and again have a 50% chance of passing. The only thing that stops it being a 75% probability is that you can't reroll a flop, so it drops to 66.7%, or two-thirds. Similarly, the percentages are better in each case if you save one die for the replace possibility.

Extra complexity

The calculation does not take account of the possibility of rolling zero successes without rolling any blanks (ie combinations of positives and negatives), but this would only increase the percentage chance of passing the challenge. It does also not show that the fewer dice you roll in the first place, the higher the probability of flopping is, and if you flop and have one momentum left then the opponent gets it. Flopping a dash is also slightly worse than failing it, as you become dazed instead of down. On the other hand, if you roll fewer dice and pass first time, then you have one momentum left for the main challenge. Happy days!

Example 3 - Dash Strategy (Part 2)

Another way that they dash can be used is to add an extra die to a challenge. Seatbelts on, this is where it gets tricky. In order for this to be worthwhile, you first need to pass (i.e. get at least one success on) the dash challenge, and then get the required number of successes on the main challenge, be it a tackle, shove, pass, whatever. Therefore the probabilities below reflect the probability of passing the dash and then getting the right number of successes with the increased dice. I'll show you the numbers, then give an example:

Extra_die_dashing.jpg
OK, here goes. You need to get at least 3 successes on a tackle. You have 4 momentum, and have a TACKLE of 2. What should you do? To make the dash worthwhile, the percentage success rate has to be higher than the chance of succeeding without the dash, which here is 13.9% (2 dice, 3 successes). You have 4 dash dice, and because you are wise you roll 3 of them, saving 1 for the replace. You have to pass the dash, which has a probability of 80%, and then get 3 successes on 3 dice, which has a probability of 28%. The overall probability of success is therefore 22%, almost double that without the dash.

At this, point, a caveat occurs. The above should be measured against the chance of getting the number of successes if you have a momentum for the challenge itself.

Example 4 - The Importance of Abilities

There are 5 fundamental Abilities dealt with here, each of which allows you to replace one dice when you do a challenge using that characteristic:
Characteristic
JOG
MIGHT
TACKLE
DODGE
SKILL
GRIT
Ability
Steady
Buff
Grapple
Nimble
Agile
Resilient

These are very powerful, particularly as they are the only way you are allowed to replace dice when you flop a challenge, and also because they are used before momentum. They massively reduce the probability of flopping a challenge using that ability, as shown:

Dice Rolled
1
2
3
4
5
6
% Flops Saved by Skill
83.3%
66.7%
60.0%
56.0%
53.2%
51.2%
% Flops without skill
16.7%
13.9%
11.6%
9.6%
8.1%
6.7%
% Flops with skill
2.8%
4.6%
4.6%
4.2%
3.8%
3.3%

The % flops saved by skill is there to aid understanding, for example if you roll one challenge die, the skill allows you to replace a flop roll. To flop the challenge, you would need to roll another flop (the infamous snake-eyes). The overall chance of flopping is therefore only 2.8%, which is equivalent to 1 in 36. Amazingly however, the challenge shows that if you have a skill, rolling 1 dice is actually the surest way of avoiding a flop. Unfortunately, avoiding a flop is not normally enough, and you want to get successes, which is easier with more dice.

In summary, if you are playing a tournament or league, with skills involved, taking the replace skills is an excellent idea, particularly on challenges you expect that player to do often, and do not want to flop.