# Single-Dice Probability in Slice & Dice

I’ve been playing a lot of Slice & Dice lately, a game where all of the heroes and enemies are defined by a 6-sided dice, a health total, and sometimes a spell or passive ability. Your party is made up of five heroes, each with a general role, who each roll their dice every turn, and the player is given the option to reroll up to two times (or more, or less, depending on other modifiers). This decision whether to keep or reroll each dice is the core gameplay element in Slice & Dice, and the focus of player decisions. Deciding what to do with each of your dice while dealing with the enemy actions and keeping your party alive for the next fight becomes increasingly complex despite the straightforward mechanics.

After playing for a while, I wanted to check my intuitions about what the best course of action in certain scenarios was, so I did what I will always inevitably do if allowed to play a game long enough: I opened up Excel and built a spreadsheet.

The following table displays the probability of getting any result on a dice given an allowed number of rerolls. Decisions are much more complicated than this — as you often need to factor in the potential results of other dice to achieve certain goals during a turn (like defeating an enemy or saving an ally), and the decision tree for even a single dice often changes depending on what the results of the the subsequent rolls become — but I’ve found this table to be a great tool for analyzing situations and making better, more informed, decisions.

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##### To use this table:
1. Determine the desired outcome, and the number of die faces that satisfy that condition. Find the number under the Target Faces header that equals this number.
2. See how many rolls you have left to achieve this outcome, and find that number under the Rolls header.
3. Find the probability listed at the intersection of your Target Faces Row and Rolls Column. This probability is the percent chance that you will see one of the target faces if you roll the dice the number of times chosen in Step 2.
##### An Example:

Let’s say that the Thief needs to roll at least 1 Damage to kill the last enemy and win the floor. The turn’s initial roll turns up the Shield 1 face, and there are two rerolls remaining for that turn.

There are four faces on the Thief’s dice that will achieve our goal of doing at least 1 Damage (the two 2 Damage faces and two 1 Damage (Ranged) faces), so our Target Faces is 4. We have two rerolls to use so our Rolls is 2. Following the 4 row under Target Faces and the 2 column under Rolls, we determine that there is an 89% Chance that the Thief can do at least 1 damage with their remaining 2 rerolls.

##### Multiple Dice

This table is really flexible. One more example of a way I’ll commonly use this is to determine the probability of two different dice achieving certain results. This is simple, you only need to multiply the individual chances of each dice rolling their respective sides. So if you have 2 rerolls remaining, your Thief needs to deal 2 Damage (2 Target Faces, a 56% chance), and your Adept needs to generate 1 Mana (4 Target Faces, and 89% chance), then you multiply the two probabilities together (in their decimal form) to get the resulting probability: 0.56 x 0.89 = 0.50, or a 50% chance.