009 - The Maths of Malifaux

The average gamer is very used to rolling dice. I would even say that most gamers develop a gut instinct for the mathematics behind the action, even if they don't view it as actually "doing maths". Forgiving the GW references, but we know that a 4+ save gives our models a 50% chance of surviving, we can feel how much better a Space Marine is at shooting than an Imperial Guardsman (2/3 vs 1/2), and we can usually quickly assess the likely result of throwing as many as 10-20 dice at a time.

By replacing dice for cards though, Malifaux puts us on the wrong foot, and it can be difficult for people to get their heads around the probabilities involved. It doesn't help that card mathematics are more complicated than dice. I'm not saying you need to be a maths genius (in fact I won't be using any techniques in this post beyond what you'll find in the average GCSE course), but it would take a very fast brain (or Rain Man!) to be able to crunch the numbers at the tabletop.

In this post I'll be:
a) Briefly explaining the maths behind the basic flips
b) Working out the probabilities of common flips and giving you the results
c) Drawing some conclusions to help build some strategies

Maths and Fate
The basic principles of card maths are easy. There are 54 cards in a Fate Deck, including Jokers, so you have a 1/54 (1.8%) chance of flipping any particular card. There are 13 cards of each suit, so if you're aiming for a trigger, your probability of flipping a specific suit is 14/54, or 25.9% (since the Red Joker can also be used as any suit you choose). If you need to flip, say, a 10 or higher to beat a duel total, your probability of success is 17/54 (31.5%). If you want both a 10+ and a specific suit, your chances plummet to 5/54 (9.3%). The general rule is that you count how many cards in the deck give you the required result, and then divide by the total number of cards, to give you the probability of success.

So far so good, hope you haven't zoned out just yet! The complication arises when you flip multiple cards together. In Malifaux, a positive flip means you flip 2 cards and take the highest. For a negative flip, you flip 2 cards and are forced to take the lowest. There are even situations where you can must make double, or even triple, positive or negative flips. If any of the flipped cards is a Joker, it wins out over the other cards, and Black Joker always takes precedence over Red Joker if you flip both.

All these rules when taken together make for some intricate combinatorics to work out the probabilities. As an example, the probability of flipping 11+ on a single negative flip is 8.2%. Any combination of two 11s, 12s or 13s works (there are 132 combinations) and any card except Black Joker, when paired with Red Joker, will also be enough to pass the duel (another 104 combinations). This means 236 out of 2862 total possible 2-card pairings will yield 10+, giving the probability 8.2%.

Bringing it all together - The Results
Hopefully I didn't leave too many people behind there! Being the maths geek that I am, I've done the heavy lifting for you and put the results into a handy reference table. Simply look for your required score in the left column (don't forget to subtract any contributing Statistic value first!) and find the probability of achieving your target score.

The below tables also summarise how much effect a positive or negative flip will have, compared to just a straight flip.

Conclusions
As you may have noticed, some interesting points can be drawn from the tables

1. Bear in mind that these are pure numbers from a shuffled deck and should be used as guidelines. The actual probabilities mid-game will also depend on what cards have already been flipped, and those in your control hand.
   
2. You may not have realised but most of the time, a double postive flip is not twice as good as a single positive flip, and the same for negatives. If your target number is high, an extra positive flip is going to almost double your changes of success, but for low/mid-range targets, it's effect is almost negligible! For negative flips the opposite is true, that is, if your target number is high, a single negative flip will really ruin your day, whereas a double negative is pretty much no worse.
   
3. If you desperately need the Red Joker to come out (scores 14), and nothing else will do, it's actually a good tactic for you to cheat down to a negative flip to increase your chance of finding it. This is rather unintuitive!
   
4. Likewise, flipping extra positive cards when you don't need to (if the duel is going to be really easy) only increases your chance of finding the Black Joker. Don't do it!
   
5. Hard to Wound models will get hit by Red Joker far more often than models without that trait. They'll also suffer more Black Jokers in the long run, but that's no consolation on the occasion when Teddy gets one-shotted by a Red Joker when your opponent is flipping a double negative for damage!
   
6. If you're up against Hard to Wound 1, a possible tactic is cheating your duel down to give you an extra negative card. This increases your chance to flip Severe damage, but decreases your chance of inflicting Moderate. Since you're already likely to do Weak damage, it's a gamble that might pay off occasionally!
   

I hope all this was interesting or useful! If you have any ideas for other ways to use the numbers I've presented here I'd be glad to hear them, and any feedback at all is very welcome!

Until next time, may your flips always be high!