Thursday, February 4, 2010

Simulating Fizzle Rates with Dice

Wow . . . I got a very cool e-mail from Benjamin Fairydust, Magus Theurgist. In this e-mail he details how to simulate fizzle rates with the roll of two dice. I think I may use this for the advanced version of the game I'm making and point to this thread for more details on how fizzle rates are simulated with the dice.

Ok, so check out this email.

~~~~~~begin cut and paste~~~~~~~~

I really like your TCG idea and was thinking of making my own deck. I recall someone’s comment regarding adding fizzling to the TCG with dice. Well after a semester or 2 of Statistics at school I’ve figured out a way to add fizzling to the game using only 2 die.

I’ve color-coordinated the TCG Fizzle Guide for simplicity: Red (NOTE: I went with a salmon color to make it easier to read on this blog) is all the mathematical and technical stuff, Blue is all the semi-technical stuff, and Green is all you really need to know (It's at the end).

This is the mathematics of it all.

If you just want how 2 die translates in fizzle accuracy skip to end:

So when you roll 2 die and take their sum (add both die together) you can get any number between 2-12 inclusive (including 2 and 12). There are 36 different combinations of numbers you could possibly get. And the chance (probability) of getting each sum (the #’s b/w 2-12) is different for each.

Below is a chart showing the sum of 2 die, the probability of each sum when both die are rolled, the different ways to get each sum, and the number of ways to get each sum:

To find the probability of each sum I simply took all the different ways of getting that sum and divided it by 36, the total number of combinations for the 2 die.

Now that we know the chance of getting each number we can use those probability to determine the accuracy for each card.

The following chart shows the accuracy of fizzling we want, the sums that get you that accuracy, and the accuracy we actually get.

So for example:

I play a Seraph card that has 90% accuracy. That means I have a 10% chance of fizzling. So I roll my 2 die.

According to the chart, we want 10% and the closest we can get is 11%. So when I roll, if I get a sum of 5 (That’s a 1,4; 2,3; 3,2; or 4,1) I fizzle.

This works for every accuracy. Just figure out what your chance of fizzling is, find the sums, roll the die, and if you get the sums listed you fizzle.

You can also do it vice versa, where the chart shows the accuracy instead of the fizzle.

Sticking to the Seraph example: Accuracy = 90% closest on chart is 91% so if we get any number that is not (3, 4, 5, 6, 7, 8, 9, 10, 12) then you fizzle.

I find using the chart to represent fizzle rate easier, but you might think differently.

For simplicity’s sake here’s a chart with the most common fizzle rates for each school:

The Fizzle #’s or Unlucky #’s (whichever you prefer) are the sums that if you get, you fizzle. So if I play a Skeletal Pirate (Accuracy = 85%), the fizzle chance is 15% so if I roll both die and they add up to 7, I fizzle.

Hope this is of some use to you.


whoa! Nice. That sounds solid enough to me! Any math people out there want to back Benjamin up? Those 3, 5, 6, and 7's are so DREADED! :-)

Happy Dueling!


Grayson said...

Ok, all I could understand is the green part. :)

Tipa said...

The math is sound, but it is way too complex to actually use.

Better to just make it a throw of one die. 1=fizzle on spirit schools, 1,2=fizzle on elemental schools, and balance damage appropriately. Or something along those lines. Simplicity rules.

Sierra Starsong said...

Maybe I've played too much D&D, but my first thought is a D20 (twenty-sided die). There's a 5% chance of any given number coming up, so you'd fizzle on:

Life - 19 or higher
Death/Balance - 18 or higher
Myth/Ice - 17 or higher
Fire - 16 or higher
Storm - 15 or higher

You could even simulate +accuracy gear with minuses to the die roll, -1 for every 5% improvement. (Rolling a natural 20 always fizzles, so there's always at least a 5% failure chance.)

On the other hand, Tipa's got an excellent point, the simpler the better. Especially if you're aiming for younger players.

And there's still pips/power pips to consider...

Fred Firepants said...

I quote what Grayson states. XD But it's very kewl :)

Benjamin Fairydust said...

Wow, that was fast, didn't think this would make it as a blog post so quickly. So let me make some clarifications:
@ Tipa
Of course the math isn't simple it's AP Statistics, but I'm trying to simulate the fizzle accuracy's as close to the actual fizzle rates in game. So the one die idea is good but not realistic.
@ Sierra
I wish I had a 20 sided die, but really how many people have one of those these days? (The answer is very few) And along the lines of keeping it realistic, I’m not sure how the math for a 20 sided die works out...hmm.
@ Everyone Else in the Spiral
Ok so I just sent all the math and techy stuff to Thomas so that he could see the ‘method to my madness’. In practicality (that means when you’re playing your TCG), all you need is 2 die(which are really easy to get and you can even take them from other board games you happen to have lying around) and a piece of paper with the following written on it:
Life=[5], Myth=[3,6], Death=[7], Fire=[5,6], Ice=[3,6], Storm=[6,7], and Balance=[7].
You play a card from a certain school, then roll the 2 die and add up the numbers, the answer you get is called the sum.(Good Math practice for kids playing the game, btw) Then you look at the Unlucky Numbers for each school on your piece of paper. If the sum is equal to the unlucky number(s) of the corresponding school, then you fizzle. Get it yet? No?
For example:
I play a Fire card, roll my 2 die and get a 1 on the 1st die and a 4 on the 2nd die. That adds up to 5, so I check my Unlucky #’s for Fire, and they’re 5 & 6, so since I have a 5, I fizzle (Fizzled cards are reshuffled into your deck).
Sounds pretty simple to me…Does that clear things up?
So all you need is 2 die, and a piece of paper with this scribbled on it:

Unlucky #’s (If I get them, I fizzle)
Life=[5], Myth=[3,6], Death=[7], Fire=[5,6], Ice=[3,6], Storm=[6,7], and Balance=[7].

Got it? Good.

Tesh said...

The math looks solid from here. It's simple enough to work (and really, if you're playing this game, a simple two-die sum is plenty easy), and simulates the percentages well enough.

I'd probably go with a d20, but I'm a gaming nerd and have several lying around. It's actually easier for me to find one of those or even 2 d10s than it would be to find to six siders. (Yes, it's odd.)

Stephanie said...

I understand it pretty much except for the second graph with the "percent we get" and "percent we want" lol

However, that is a very interesting way to look at the fizzle rates. If it was me, I would just simply pick 1-storm, 2-ice, 3-fire, 4-balance, ...ect. That way, if I wanna deal the card but when I roll the "die", and it landed on that number, then I fizzle.

Morgrim said...

Sure, the math is sound, but a 20-sided die (d20) is maybe 50 cents, and that's what I would use regardless of what's in the rules. maybe both options could be included.

This is a TCG, and many players or former players of the first TCG, Magic: the Gathering, have a d20 or two that they use(d) as a life counter for that game.

A d20 is an icosohedron, and it's 20 faces are all equilateral triangles with the same area, so each has a 5% chance to come up. Since it's more intuitive to have higher numbers being better, you'd want the roll to represent the fizzle rate rather than the accuracy. I'm not sure... is there a cap in the game that keeps you from getting above 95% or below 5% accuracy? If so, rolling a 1 always fizzles and rolling a 20 always succeeds.

On a d20:
School____Roll Above
Life______________ 2
Balance, Death____ 3
Ice, Myth_________ 4
Fire______________ 5
Storm_____________ 6

Multiply the "Roll Above" by 5, and that's your odds of fizzling. Perfectly simple and perfectly accurate. If the were licensed by KingsIsle to be sold in stores, the boxes containing starter sets could easily be made a bit longer and include a couple d20. you could differently colored ones for each school, using the school colors. Black with white numbers for Death, dark green with light tan numbers for Life, and so on.

I was also thinking, in addition to spell cards, there could be some cards that represent equipment. Say each player gets 10 points (call them Crowns) worth of equipment. Different items have different values, and you cannot use more than 1 equipment card of the same type (like 2 robes at once). You select the items you want to use before the duel and lay them face up next to your play area for easy reference. You can't change equipment during a duel. There could be a robe that reduces the fizzle rate of a specific school by 5% (subtract 1 from the "roll above" number), A hat that does the same for Power Pip chances, and shoes that add 10 bonus health. Storm deck players would likely choose as much accuracy as they could, while Ice might want damage boosts. Instead of handling damage increases and resists as percentages, just stick to straight integers to be added or subtracted. Keep the numbers low, more similar to equipment from Unicorn Way than Dragonspyre. Keep percentages in increments of 5%, and keep damage/resist adjustments in the 1-10 range.

Anonymous said...

A custom D20 would do the trick nicely.

2 faces for life (19,20)
3 faces for death and balance (16,17,18)
4 faces for ice and myth (12,13,14,15)
5 faces for fire (7,8,9,10,11)
6 faces for storm (1,2,3,4,5,6)

Should KI make a table top version this would be perfect, if your school comes up you fizzle.

as for 2d6 system, its flawed
death and balance get a -1.7%
life gets -1.1%
storm -0.6%
fire +-0%
and ice and myth get +0.6% bonus.
this is the deviation from the % printed on the card.

go with a d20 or even 2d10. the math is pure that way. (although i suppose you could do it cleanly with a d4 and a d5 but who has d5's anyways... yea i want 1).

panglou said...

Eeek! I need a calculator, lol, I can barley undestand, but I get it... Kinda. ;)

Jason Dragonblood said...

Just use 2 D10's.
I have a ton of them some are 0-9 and some are 1-10. I would use 2 of the 0-9's one blue (for the first number) and a red one (for the second number.

so if you roll 0 on the blue and 1 on the red it would be 01.

5 on blue and 4 on red would be 54

if you roll double 0's that would be a 100.

Using this method if life cards are 90% cast rate the card could be played as long as you roll a 90 or under on the dice. A 91-100 would be a fizzle.

stingite said...

Thanks for the feedback all! I definitely would love to keep this all simple . . . hmmmm.

AND, either way, thanks Benjamin for your time thinking about this. I appreciate it. :-)

Happy Dueling!