Multi-Attack Probability Tree

I have been trying to figure out good non-sandstorm pets to use against Anomalus. With Anomalus only having 1034 health I have been using the strategy of trying to take the annoying little blob out as fast as possible, before it can get off a lot of attacks. One pet that I have used somewhat successfully is the S/S Flayer Youngling. At 325 speed it is faster than Anomalus. If you use the sequence Kick -> Blitz -> Blitz there is a chance that you will take out Anomalus on round three, only allowing it one attack on round two (since it is interrupted by Kick on round one and the Flayer Youngling goes before Anomalus on round three). What makes this happen is the multi-attack Blitz hitting three times on round two and round three.

While making videos I have stated a number of times that the probability of killing Anomalus on round three is 25%, which is the probability of Blitz landing its third attack on both round two and round three (assuming the probability of landing each Blitz’s third attack is 50%). I realized at some point that this is not entirely accurate. Critical hits are not taken into account. I set out to find the true probability of killing Anomalus on round three, which meant that I had to take the chance of landing critical hits into account. When when an attack crits, it does 150% (or 3/2) damage.

Keep in mind that these calculations are assuming that the opponent starts with Anomalus, then stays in versus your Flayer Youngling. This is PvP and you never know what your opponent will do, but I’ve found that most Anomalus users start with it and often don’t swap it out even when it would be a good idea to do so.

Note: right after I typed this I faced an Anomalus with my Crystal Spider (discussed below), and after getting beaten in a few previous battles by starting with Anomalus versus Crystal Spider the opposing player started with Graves instead, immediately contradicting what I had just typed haha. However, I sill won because they were forced to alter their strategy :)

I make the two assumptions that landing the third attack on a multi-attack while going first is 50% and the probability of getting a critical hit is 5%. If either of these assumptions are wrong a few numbers will have to be changed, but the process will still hold up.

Below is the probability tree that I made for one of Blitz’s attacks. Rather than use the number 140 for Blitz’s damage I decided to generalize it by using a variable, x, for the damage done by one attack. That way it can be easily applied to other pets. Here are the symbols used in the tree:

x = a variable representing damage from one attack
N = normal hit
C = critical hit

So the numbers next to each branch of the tree represent the probability of that event happening, with the damage next to it (x for N, 3/2x for C). The variables are added up for each series of events at the bottom, along with the probability calculations found by multiplying the probabilities on each of the branches. Those products are boxed. The third row of the tree represents whether or not the multi-attack hits or misses its third attack, the other rows represent a normal or critical hit.

IMG_4793

The next step was to add up the probabilities for each of the damage totals.

IMG_4787

Note: from this point on I switched from using fractions to using decimals for presentation purposes.

P(2x) = 7220/16000 = 45.12%
P(2.5x) = 760/16000 = 4.75%
P(3x) = 6879/16000 = 42.99%
P(3.5x) = 1083/16000 = 6.77%
P(4x) = 57/16000 = 0.36%
P(4.5x) = 1/16000 = 0.006%

To be clear how these numbers were found take a look a 4x.There are three ways that you can get 4x damage: normal crit crit (NCC), crit normal crit (CNC) and crit crit normal (CNC). Since there are three ways this can happen and the probability of each is 19/16000, you get P(4x) = (19 + 19 + 19)/16000.

You would use this information as follows. Say you want to know the probability of the Flayer Youngling doing 420 damage with a Blitz. 420 = 3*140, so you would look at P(3x) = 42.99%.

Here’s the same info as above but with x = 140, so you can see Blitz’s damage from an S/S Flayer Youngling.

P(2x) = 7220/16000 = 45.12% (280 damage)
P(2.5x) = 760/16000 = 4.75% (350 damage)
P(3x) = 6879/16000 = 42.99% (420 damage)
P(3.5x) = 1083/16000 = 6.77% (490 damage)
P(4x) = 57/16000 = 0.36% (560 damage)
P(4.5x) = 1/16000 = 0.006% (630 damage)

Ok, now that that part was done it was time to look at the Flayer Youngling versus Anomalus. Anomalus starts out with 1034 health, the flayer is going to start by using Kick.

Kick can either do 210 damage or 315 damage with a crit.

1034 – 210 = 824
1034 – 315 = 719

Assume Anomalus got hit with a normal Kick and has 824 health. 140*6 = 840 , so we need to do at least 6x damage over the next two attacks.

Rather than find the probability of the flayer killing Anomalus by finding all the ways it can do at least 6x damage, it was easier to find the probability of the flayer NOT killing Anomalus, so I chose to look at the combinations of two Blitz’s that led to damage less than 6x (840).

Those combinations are:
2x + 2x = 560 damage (1 way)
2x + 2.5x = 630 damage (2 ways)
2x + 3x = 700 damage (2 ways)
2x + 3.5x = 770 damage (2 ways)

2.5x + 2.5x = 700 damage (1 way)
2.5x + 3x = 770 damage (2 ways)

“2 ways” refers to the fact that you can get the result in either order, for example you can to 5x damage by doing 2x then 3x or by doing 3x then 2x.

P(not killing Anomalus after a normal Kick and 2 Blitzes) =
[P(2x)*P(2x) + 2*P(2x)*P(2.5x) + 2*P(2x)*P(3x) + 2*P(2x)*(3.5x) + P(2.5x)*P(2.5x) + 2*P(2.5x)*P(3x)] / [16000^2] = 0.7387

Therefore P(killing Anomalus after a normal Kick and 2 Blitzes) = 1 – 0.7387 = 0.2613

Now assume Anomalus got hit with a critical Kick and has 719 health. 140*(5.5) = 770, so we need to 11/2x damage over the next two attacks.

Again, rather than find the probability of the flayer killing Anomalus, there were less ways of the flayer NOT killing it, so I chose to look at the combinations of two Blitz’s that led to damage less than 5.5x.

Those combinations are:
2x + 2x (1 way)
2x + 2.5x (2 ways)
2x + 3x (2 ways)

5/2x + 5/2x (1 way)

P(not killing Anomalus after a critical Kick and 2 Blitzes) =
[P(2x)*P(2x) + 2*P(2x)*P(2.5x) + 2*P(2x)*P(3x) + P(2.5x)*P(5/2x)] / [16000^2] = 0.6368

Therefore P(killing Anomalus after a critical Kick and 2 Blitzes) = 1 – 0.6368 = 0.3632

Ok, the final step. Now take the probability of a normal Kick times the probability of killing Anomalus after a normal Kick, then add that to the probability of a critical Kick times the probability of killing Anomalus after a critical Kick.

P(killing Anomalus on round three) = (0.95)(0.2613) + (0.05)(0.6368) = 0.2664.

So there is a 26.64% that the Flayer Youngling will kill Anomalus on round three, allowing Anomalus only one attack. Phew, those crits made life difficult.

But wait, there’s more!

I was curious about the Scalded Basilisk Hatchling (SBH) versus Anomalus. After all, the S/S Bronze Whelpling is probably the BEST pet one on one starting against Anomalus. Crystal Prison -> Arcane Slash x 3 (350 + 350 + 362 damage) is guaranteed (100% chance) to allow Anomalus only one attack. I’ve always thought of the Bronze Whelpling as a lesser version of the almighty SBH, but it turns out in this instance that’s not the case. The reason is because the Bronze Whelpling doesn’t rely on multi-attacks, Arcane Slash gets its extra damage from going first, which it will because it’s faster than Anomalus. The SBH, however, relies on multi-attack rng for its extra damage.

The SBH will Crystal Prison -> Thrash -> Thrash -> Thrash -> Thrash. A Thrash hit does 140 damage. In order to kill Anomalus, the SBH needs to do (15/2)*140 = 1050 damage, or 15/2x. Interestingly, there is a possibility that the SBH will kill anomalus on round three, by doing 15/2x damage with two Thrashes.

These combinations are:

4.5x + 4.5x (1 way)
4.5x + 4x (2 ways)
4.5x + 3.5x (2 ways)
4.5x + 3x (2ways)

4x + 4x (1 way)
4x + 3.5x (2 ways)

P(SBH killing Anomalus on round three) =
[P(4.5x)*P(4.5x) + 2*P(4.5x)*P(4x) + 2*P(4.5x)*P(3.5x) + 2*P(4.5x)*P(3x) + + P(4x)*P(4x) + P(4x)*P(3.5x)] / [16000^2] = 0.0006 = 0.06%

Umm, good luck with that. Remember that 4.5x represents three crits, so it makes sense this probability would be low.

Ok, how about round four after three Thrashes? Once again, it’s easier to find the probability that the SBH doesn’t kill Anomalus with three Thrashes. Here are the ways three Thrashes end up below 15/2x:

2x + 2x + 2x (1 way)
2x + 2x + 2.5x (3 ways)
2x + 2.5x + 2.5x (3 ways)
2x + 2x + 4x (3 ways)

P(SBH doesn’t kill Anomalus on round four after three Thrashes) =
[P(2x)*P(2x)*P(2x) + 3*P(2x)*P(2x)*P(2.5x) + 3*P(2x)*P(2.5x)*P(2.5x) + 3*P(2x)*P(2x)*P(4x)] / [16000^3] = 0.3866

Therefore,
P(SBH kills Anomalus on round four after three Thrashes) =
1 – P(SBH killed Anomalus on round three) – P(SBH didn’t kill Anomalus on round 4) = 1 – 0.0006 – 0.3866 = 0.6128

That’s all the probability for now, but there are some more pets I wanna check out. Those involving high/low attacks will be difficult. I did figure out the high/low probability distribution before (Burn Damage Distribution) so it can be done, but I’m not feeling it now (wasn’t really planning on writing this up >.<). It would require some coding for sure. Some Anomalus killers can be found with simple addition. For example, the Mongoose Pup can Gnaw x 3 (350*3 = 1050) to allow Anomalus only two attacks. The Crystal Spider can Crystal Prison -> Web-> Spiderling Swarm -> Web to allow only one Anomalus attack, while Dandelion Frolicker can Kick -> Scratch x 3 to allow only two attacks. Also, Whomper’s Flash Freeze -> Deep Freeze -> Punch -> Punch will keep Anomalus at one attack. There are a lot of Scorched Earth, Breath and Flame Breath pets I want to look at as well.

Here are the results I’ve come up with so far (edited on 8/13, thanks Drevoed (see comments)). As mentioned above, this is based on a beginning game showdown with Anomalus not swapping out.

Pets that allow Anomalus zero attacks:
SBH 0.06%

Pets that allow Anomalus only one attack:
Bronze Whelpling 100%
Crystal Spider 100%
Whomper 100%
SBH 61.28%
Flayer Youngling 26.64%

Pets that allow Anomalus only two attacks:
SBH 100%
Flayer Youngling 100%
Mongoose Pup 100%
Dandelion Frolicker 100%

And finally, I wrote some code to double check myself and ended up with the same results. This doesn’t necessarily help prevent logical errors (because if I made a logical error computing by hand then I made the same error in coding), but it did help to verify the numbers. In fact, I computed the SBH probabilities wrong by leaving out a 760 somewhere, then got a different result when I coded, which forced me to find the error.

One of the calculations is the expected value of a multi-attack going first, which is 2.54563x. So in the case of Blitz or Thrash that do 140 damage per attack, the expected value of the damage is 140*2.54563 = 356.4

Here’s a link to the code. The first part is just the calculations, below that are the loops used to verify the probabilities.

Mathematica code

12 Responses to Multi-Attack Probability Tree

  1. Drevoed says:

    I think you meant to say SBH has 0.06% to let Anomalus zero attacks, 61.28% to let only one attack and 100% to let only 2 attacks, not 0.06% to let one and 61.28% to let two.

    • Discodoggy says:

      I think after I find some more pets that take out Anomalus quickly a round by round table would be nice, showing the moves of each pet.

  2. Discodoggy says:

    Thanks for taking a close look at this. Yes, you are right.

    Round 1
    SBH Crystal Prison
    Anomalus stunned

    Round 2
    SBH Thrash
    Anomalus stunned

    Round 3
    SBH Thrash (0.06% of killing, which means Anomalus got 0 attacks)
    Anomalus attacks if alive

    Round 4
    SBH Thrash (61.28% of killing, which means Anomalus got 1 attack)
    Anomalus attacks if alive

  3. Drevoed says:

    SS */2/2 Sneaky Marmot allows Anomalus only 2 attacks with a good chance one or both of those attacks will be his no-cd first abilities thanks to blind / Smoke bomb canceling aoe.

    • Discodoggy says:

      Nice! I haven’t used the marmot outside of Murkalot counters.

      Also, I pretty much gave forgot about this little project for now. School started and distance learning is sucking my soul dry. But I appreciate you still adding to the list.

  4. Noel says:

    Hey Disco, I’m working on an app and would love your input; its a live view of your pets, with filtering etc Anyway I’ll send you more info, can you email me I’ll send you some pictures.

    I realize you’re busy with school, distance learning etc so I promise to demand nothing lol

  5. Drevoed says:

    Did you guys know that Crispin with Nut Barrage and Adrenaline Rush only lets Anomalus do two attacks 73% of the times?

    Chance of hitting six out of six 80% to hit nuts: (4 / 5)^6 = 26.2%
    Chance of hitting five out of six 80% to hit nuts: 6 * (4 / 5)^5 * (1 / 5) = 39.3%
    Chance of hitting four out of six 80% to hit nuts plus critting with anything out of 7 attacks (1 adrenaline rush, 4 nuts, 2 dot nuts, yep, even their crit is enough): 6 * 5 / 2 * (4 / 5)^4 * (1 / 5)^2 * (1 – 0.95^7) = 7.5%
    Total: 26.2% + 39.3% + 7.5% = 73%
    Not counting miniscule chance of hitting 3 nuts with multiple crits.

    It’s the most prominent example of crits mattering I could find. Swinging from slightly worse than 2 in 3 odds to slightly worse than 3 in 4!

    • Scyre says:

      Anomalus is so annoying. Since it has been introduced I feel I need to run teams all the time that protect for it (and I do not enjoy that). I am finding a number of pets that do well against it. I agree with Sneaky Marmot and the basilisk (although the basilisk is lower end). Thanks for the tip on Crispin, I am going to start making teams with it now. Below is a list of others (of course the all assume there is no swap in the mix):
      SS Bronze Whelpling (prison, tail sweep tail sweep) guaranteed 1 attack only but a risk of swap or a smarter player will see the strong to magic and not start with anom.
      SS Qiraji: (Blackout, Crush, Crush, Sandstorm) where there is a chance of only 1 attack, but even if there are two the sandstorm limits the dot damage significantly (this is by far my fav).
      HS Crystal Spider where the key is to use Sticky web first to prevent that swap and then you use the prison, swarm, web which only allows 1 attack. When this works it’s great. the downside is the cooldowns.
      Kunchong where you use Poison fang and puncture wound which allows 2 attacks with this so carry a heal pet along with it.
      SS Shadow Back Crawler works and you carry darkness pets with it. Be sure to use darkness as the killing attack to ensure a kill.
      Another combo I like to use is two swap pets that are faster than anom, this really messes the sequence up.

  6. K says:

    3 spiders should do the job. Crystal Spider is very good for the crystal prison, but spiders faster than 297 using brittle webbing and spiderling swarm kill Anomalus pretty quickly.

  7. Mutanis says:

    Another good pet that allows Anomalus only one attack is P/S Rose Taipan (1/1/2)
    1 blinding poison –> 2 poison fang –> 3 puncture wound –> 4 poison fang (Anomalus dies)
    r1: 0
    r2: 154 and 77
    r3: 309 + 309 + 77
    r4: 154
    (1080 dmg)

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