[Paladin] Math

During a dull moment at work I tried to calculate how Paladin's dice mechanics work.

Essentially each dice in a pool has an expected value of 0.5. You can spend a point of Animus to increase the size of the pool. Once you've fixed the size of the pool, you then roll... and can "activate" your Animus to reroll any result under a certain value. The first of these activations is free. Yet you can continue to re-roll by spending a point of Animus each time.

There's a resource management consideration, then: do you spend your Animus points to increase the size of your pool... or do you spend those points on rerolls?

By fooling around with it through trial and error (I've forgotten all the probability theory I ever knew, alas), it looks like it's always better to increase the size of your pool. Since the first activation is free, later activations only give marginal returns.

If anyone has worked out the actual equation for this, I'd like to see it because it was driving me crazy during a dull meeting.

BTW, these are the expected values after your "free activation," depending on certain animus values:

(Animus Value, Expected Value of Each Dice)
(0, 0.5)
(1, 0.58)
(2, 0.66)
(3, 0.75)
(4, 0.83)
(An Animus attribute of 5+ is harmful)

Or.... for an Animus value X, the expected value is (6 + X)/12 on the free activation. Haven't worked it out for the others yet.