Let's say you're creating a role-playing game, and you've decided you want the base of the system to be rolling two six-sided dice (ie. 2d6). In general, you roll those dice, add them together, add some sort of modifier based on your skills or character, then compare to some target number. But let's say that you want to have a system where you make that standard roll, then after succeeding you need to flip a coin to determine some additional factor. Perhaps you roll to see if you hit an enemy, then roll to determine if they've been poisoned as well. You could do this by rolling the 2d6, then doing some other random action (rolling a die, flipping a coin etc.) to give you the secondary 1/2 probability. But in the interest of making things smoother, it'd be nice if the single random action could determine both parts of the probability. So how can we do this? If our dice are distinct (ie. one red die and one blue die) then we can do something like: P(Poison) = P(4 or greater on the red die). This nicely splits the grid into two even portions. But, if the player only has dice that match then we are relying on something like positioning (left die vs right die), or the order in which we roll the dice. Things like this, where we rely on the dice being distinct, also make it easier to cheat. Imagine "No I said the blue dice was the key one, not the red one", "No that was the die from my left hand, they crossed over", etcetera etcetera. Instead, it'd be great if we didn't have to rely on the dice being distinct. So now we're trying to partition the grid of 2d6 into two even halves, where the grid is symmetrical on the A=B axis. Let's start with a simple option: P(Poison) = P(A 1 or a 2 is rolled on at least one die). This gives us a relatively even split, it's not perfect, but it's pretty good. There is a 20/36 chance of getting poison, and a 16/36 chance of not getting poison (we're going to be leaving our probabilities over 36, rather than simplifying, just for consistency). This method is reasonably even, easy to check, but has a key flaw. This issue being, we haven't considered conditional probabilities. We're going to step back to our original context, where we roll to hit, then roll for poison. One of the key parts, is that the poison is only relevant if we already know we're going to hit. We currently have: The 2d6 dice roll, the modifier, and the target number. We're going to subtract the modifier from the target number, so that we've only got a single variable: the new (modified) target number. Let's call it X, for the sake of easy reference. Now, let's go back to the system where we are looking for a 1 or a 2. If the X is 12 we need to roll two sixes, so no poison. If X is 11, then we need to roll a five and a six, so once again, no poison. Now, if X is 10, we're in a similar situation. We either roll two fives, or a four and a six, so no poison. If X is 9, we need three and six, or four and five, neither result in poison. If X is 8, we can finally actually get poison by rolling a two and a six, but even still, the chances of rolling an 8 or greater are 15/36, only two of these rolls give poison, so there is a 2/15 chance of poison, still nowhere near the 1/2 we were looking for. We continue down the line of probabilities getting: P(Poison|X>=7) = 6/21, P(Poison|X>=6) = 10/26, P(Poison|X>=5) = 14/30, P(Poison|X>=4) = 17/33, P(Poison|X>=3) = 19/35, P(Poison|X>=2) = 20/36. So the probability of poison crosses over 1/2 between X=4 and X=5. AAAAAA AAAAAA AABBBB AABBBB AABBBB AABBBB