Sloane's impact on blue dice anti-ship is pretty straightforward. 1/4 of the initial results are useless and get rerolled, and we can quickly see that basic TIEs have only a 1/16 odds to whiff completely.
But Phantom's are another story entirely. Two dice, of which 1/4 are initially whiffs, only one of the accuracies can be spent, and only one of the crit faces can be rerolled. So, what's the impact?
Summary, comparing one Phantom to a pair of basic TIE fighters:
- 5/32 (16%) odds of a whiff, versus two TIEs at 1/256 (< 0.5%)).
- 21/128 (also 16%) odds of doing more than two TIE's can: 3-4 damage, or an accuracy and 2 damage.
- 1.22 damage average, versus basic TIE at 0.62 damage (or 1.24 as a pair). Compare that to a BCC-enabled TIE bomber at 1.25 on average.
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About the same odds as a basic TIE of generating an accuracy overall, at 28% for the Phantom to generate one, versus 31% for a single TIE fighter.
- Two TIE fighters do better, at 53% for two TIEs to generate at least one accuracy. But I think that's a less important figure, as I discuss below.
So overall, I think its not quite as good as two TIE fighters: doing about the same on average while requiring a much higher dependence on the Force for your rolls. On the flip side, its not quite as expensive in points either, and much less expensive in terms of activations. A Gozanti activating 2-3 Phantoms will have a more significant impact than activating TIE fighters, or even activating TIE bombers with BCC. I think that makes the Phantom somewhat less of the hard pass that it currently is, and makes it worth considering in some fleets. That's because the impact of accuracy generation should be measured per-activation, not per-fighter IMO, and the Phantom does about the same as a basic TIE in that regard.
To attack this problem, I needed something with more oomph than mere 2D convolutions in Matlab/Octave: I turned to the open source computer algebra system Maxima. If you've ever worked with Maple or Mathematica before, Maxima is in the same class of software, but free (both free-as-in-freedom and free-as-in-beer). 'wxmaxima' is available through many Linux systems' package manager, and 'wxMaxima' as packaged on Sourceforge can be downloaded for Windows systems.
I resolved the statistics by encoding the die results as polynomials in three variables (a = accuracy, d = damage, c = crit). Substitute c^2 results for c to capture that only one crit can be rerolled, and substitute a^2 for a to capture that only one accuracy can be spent. Then perform the reroll and accumulate the results. The final polynomials are kindof a mouthful:
TIE Phantom: (3*d^4)/128+(3*d^3)/32+(3*a*d^2)/64+(27*d^2)/128+(3*a*d)/32+(15*d)/64+(9*a)/64+5/32
2x TIE Fighters: (25*d^2)/64+(25*a*d)/64+(5*d)/64+(25*a^2)/256+(5*a)/128+1/256
I like this encoding of the problem better than using convolutions, at least as far as expressing the problem goes. I'm still learning the polynomial manipulation functions to make the results easier to read, though.
/* one red die, encoding results as d=damage, c=crit, 1=zilch, a=accuracy */ red: d/4 + d^2/8 + a/8 + c/4 + 1/4; blue: d/2 + c/4 + a/4; basic_sloane_tie: expand(subst(c=1, subst(c=blue, blue))); two_sloane_ties: expand(basic_sloane_tie*basic_sloane_tie); /* Multiple accuracies don't count, and neither do multiple crits */ phantom: subst([a^2=a, c^2=c], expand(red^2)); /* Perform the reroll of one die with a crit facing */ phantom_sloane: expand(coeff(phantom, c, 0) + coeff(phantom, c, 1)*red); /* at this point, accuracies only count once, crits cannot be rerolled */ phantom_sloane: subst([a^2=a, c=1], phantom_sloane);
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