Heroes of the Storm

The Probability Distribution of Rain of Destruction. How much does it really do?

HeroesoftheStorm 6 - The Probability Distribution of Rain of Destruction. How much does it really do?

Rain of Destruction got a massive buff last patch and is receiving a lot of attention from adventurous players willing to try it out. I'm here to go over the math of Rain of Destruction, analyze its random nature, and figure out just how much damage you can expect this ability to do.

So, how do we calculate the expected damage of Rain of Destruction? We need a few values first. We'll need to know the area of effect of the ability itself and the impact area of each individual meteor so we can calculate the probability of each meteor landing in a particular spot. Then we'll need to know how many meteors there are in total, and we have to account for the fact that larger heroes will get hit more often on average. So, after some measuring, I've determined that Rain of Destruction itself has a radius of 9.5 and each meteor has a radius of 1.25, however, if someone knows these values to be different, please correct me and I'll update my math. But then we need to account for differing hero radii. This is because abilities in HotS only need to intersect a non-zero portion of their areas with the hero to register a full hit, i.e getting hit right at the edge of the meteor AoE is the same as getting hit by the center. So if you are standing in a particular spot, the area that the center of the meteor must land in order to hit you is a circle centered on your hero with radius equal to your hero's radius + the radius of the meteor, 1.25. Now we can calculate the probability of getting hit by a meteor simply by dividing the 2 areas.

The area of a circle is pi*radius2, therefore the area of Rain of Destruction is pi*9.52, and the area each meteor must land in to register a hit is pi*(1.25+h)2 where h is your hero's radius. When we divide the second area by the first, the pi terms cancel out and we get that the probability that a meteor landing in a random spot will hit you is (1.25+h)2 /9.52. So if you had a radius of 0.75, then this would be 22 /9.52 = 4/90.25 = 4.43%. That's an unlikely event, but there are a lot of meteors. 56 in fact, raining at 8 per second for 7 seconds. With those calculations out of the way, we can get to the real meat of the problem.

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Now since every meteor either hits or does not, we can treat this problem as a binomial distribution. The formula for which is px * qn-x * nCx where p is the probability of getting hit by a meteor, q is 1-p, x is the number of meteors you get hit by, n is the total number of meteors (56), and nCx is n choose x. Multiplying all these numbers together with the above formula gives us the probability of getting hit by x meteors. By plugging different values for x and all the other constants as well as differing unit radii (r), we get the following probability distribution:

0-Hit 1-Hit 2-Hit 3-Hit 4-Hit 5-Hit 6-Hit 7-Hit 8-Hit Average
r=0.5 14.5% 28.5% 27.5% 17.4% 8.09% 2.95% 0.882% 0.221% 0.0476% 1.90
r=0.6 11.5% 25.3% 27.5% 19.5% 10.2% 4.17% 1.40% 0.0393% 0.0950% 2.12
r=0.7 8.98% 22.1% 26.7% 21.2% 12.3% 5.65% 2.11% 0.663% 0.179% 2.36
r=0.8 6.92% 18.9% 25.4% 22.4% 14.5% 7.35% 3.05% 1.06% 0.318% 2.61
r=0.9 5.26% 15.9% 23.6% 23.0% 16.4% 9.22% 4.23% 1.63% 0.539% 2.87
r=1.0 3.94% 13.1% 21.5% 22.9% 18.1% 11.2% 5.64% 2.39% 0.872% 3.14
r=1.1 2.91% 10.6% 19.1% 22.4% 19.3% 13.1% 7.25% 3.38% 1.35% 3.43
r=1.2 2.12% 8.45% 16.6% 21.2% 20.1% 14.9% 9.00% 4.58% 2.00% 3.72

There are 2 major points I want to make from this table: First, the damage that Rain of Destruction deals is hugely variable. You might get hit once, or you might get hit six times, and neither you nor Gul'dan can control that. So even if the damage were respectable, it's completely unreliable. You can never be sure that you'll secure a kill with it, even if you only need to hit the target once.

And secondly, the damage isn't respectable. Many heroes average 3 hits or less which is 480 damage over seven seconds. 480 damage is a good amount, but not when it takes 7 seconds to apply. That's only 60 dps which is literally the same dps as Tassadar's tickle beam. It's not worth it. But even if the damage were greatly buffed, you still have the problem of massive inconsistency of the heroic. While the recent change is a massive buff and greatly appreciated, it is still nowhere near enough to make the ability competitive. The ability's damage needs to be made higher and more consistent for it to have any chance of competing with Horrify.

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Edit: Something I forgot to mention, all of this math is assuming that you stand in the rain for the entire 7 seconds. If you exit the radius early, the averages drastically decrease. Admittedly it's not too hard to get people to stand in it for the whole 7 seconds, but it must be taken into account that this will not always happen.

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