In our dice pool mechanic, there are effectively three inputs:

1. Number of dice in the pool.

2. Target number.

3. Number of successes needed.

Modifiers to the dice pool are another input, but from the probability standpoint they only lower the effective target number. So in all my analysis I'll assume that the modifier is 0.

To plot this fully would be a three dimensional grid - but since holographic displays are not yet a reality, I'll be breaking this down into bit size chewy bits.

The first things to note are the obvious ones. There is no possibility of success if the target number is 7 or higher (after taking into account any modifiers). Also, there is no possibility of success if the number of required successes is greater than the size of your dice pool. These things seem obvious, but they will be serious constraints on the design of the system, because impossible tasks just aren't much fun.

(Shadowrun uses d6 dice pools, but they overcome this particular limit by using open-ended rolls - for each 6 you roll, you can roll another die and add it to the total. This way, there's no theoretical upper limit.)

So, without further ado, here's my first probability table. This one is for rolling DicePool(3,d6) against various target number, requiring only a single success. That's an easy number of successes and a moderate sized dice pool.

[table=Successes >= 1]

[tr][th]Target[/th][td]6[/td][td]5[/td][td]4[/td][td]3[/td][td]2[/td][/tr]

[tr][th]DicePool(3,d6)[/th][td]42%[/td][td]70%[/td][td]88%[/td][td]96%[/td][td]99.5%[/td][/tr]

[/table]

Obviously, with only a single success required, this isn't too hard, even with the hardest difficulty. Reducing the difficulty (or equivalently, increasing modifiers) rapidly increases the probability even more.

Now lets try that with more difficulty. We'll leave the same dice pool, but require 2 successes.

[table=Successes >= 2]

[tr][th]Target[/th][td]6[/td][td]5[/td][td]4[/td][td]3[/td][td]2[/td][/tr]

[tr][th]DicePool(3,d6)[/th][td]7.5%[/td][td]26%[/td][td]50%[/td][td]74%[/td][td]93%[/td][/tr]

[/table]

A big difference! At the high difficulty, it has gone from easy (2 out of every 5) to pretty darn hard (1 out of 12 or so). Even so, if the difficulty drops to 4 (or the attacker has a +2 modifier over the defender), the probability is still fairly high.

Lastly, try it with three required successes. That means you'd have to succeed on every die:

[table=Successes >= 3]

[tr][th]Target[/th][td]6[/td][td]5[/td][td]4[/td][td]3[/td][td]2[/td][/tr]

[tr][th]DicePool(3,d6)[/th][td]0.5%[/td][td]3.7%[/td][td]12%[/td][td]30%[/td][td]58%[/td][/tr]

[/table]

Here, even if the task is ridiculously easy (target of 2), there's still a good chance to fail. At the high end, this is down to just 1 in 200.

The take home lessons are probably these:

1. Changing the target number has a huge effect on the probabilities.

2. When dice pools are relatively small (3 in this case), changing the number of successes required also has a rather drastic effect.

I rather suspect that things would be rather different if dice pools were larger and the number of successes required were in the area of 1/4 to 1/2 of the number of dice in the pool. So later I will come back and try this again using a larger

dice pool, like 10d6, and successes ranging from 3 to 5 or so.