Improved Shadow Bolt

Effects of Improved Shadow Bolt in conjunction with Ruin

With the talent Improved Shadow Bolt there are 4 kinds of hits:

1)non-imp crit
2)non-imp norm
3)imp crit
4)imp norm

Formulas

Given p the chance to crit and a the chance for a hit that is not improved, the following formulas describe the distribution of the 4 different cases. If you had an infinite amount of mana and could cast an infinite amount of Shadow Bolts and always managed to cast 4 Shadow Bolts before the timer expired, the percentages would converge to the following:

non-improved crit	p*a
non-improved normal	(1-p)*a
improved crit	p*(1-a)
improved normal	(1-p)*(1-a)

A hit can only be non-improved if the previous 4 hits were non-crits, which give

a=(1-p)^4

Distribution

So, for the following rates you get the following distribution between the different kinds of hits (based on the above formulas):

crit-rate	non-imp-crit	non-imp-norm	imp-crit	imp-norm
0.0%	0.0%	100.0%	0.0%	0.0%
5.0%	4.1%	77.4%	0.9%	17.6%
10%	6.6%	59.0%	3.4%	31.0%
15%	7.8%	44.4%	7.2%	40.6%
20%	8.2%	32.8%	11.8%	47.2%
25%	7.9%	23.7%	17.1%	51.3%
30%	7.2%	16.8%	22.8%	53.2%
100%	0.0%	0.0%	100.0%	0.0%

Note that the higher the crit rating, the more Shadow Bolts will be affected by Improved Shadow Bolt, which can lead to an overall increase in damage depending on the chance to crit.

Value of crit with Improved Shadow Bolt

The added value to Shadow Bolt with [Ruin] is 100% for crit normal, 0% for non-crit normal, 120% for crit improved and 20% for non-crit improved. Without [Ruin], you get 50%, 0%, 80% (1.2*1.5-1) and again 20%. The average value of a Shadow Bolt with Ruin will be the sum of probabilities from the distribution each multiplied with the added vale:

ap*1 + (1-p)a*0 + p*(1-a)*1.2 + (1-a)(1-p)*0.2

Without [Ruin] you get:

ap*0.5 + (1-p)a*0 + p*(1-a)*0.8 + (1-a)(1-p)*0.2

With the following crit rates you get

Damage increase versus crit rate

crit rate	w ruin	w/o ruin
0%	0.0%	0.0%
5%	8.7%	6.3%
10%	16.9%	12.2%
15%	24.6%	17.8%
20%	31.8%	23.0%
25%	38.7%	27.9%
30%	45.2%	32.5%
100%	120%	80.0%

Summary

Improved Shadow Bolt with [Ruin] seems to work very well together, getting an extra 30% or more damage with 20% or better crit rate, which is a realistic goal. The other point here would be that Shadow Bolt should scale very nicely with levels and gear if Improved Shadow Bolt and [Ruin] are taken.

After about a 20% crit rate is attained, the returns on crit rate seem to be diminishing with respect to Improved Shadow Bolt, but on the other hand it looks like [Ruin] makes up for this according to the damage table.