I needed to update my thread with some relevant data regarding eternal pets. I have posted some of the information below in another section of the forum but that information will be lost.
Discussion and figures related to obtaining a perfect eternal pet.
A perfect eternal pet is a pet that will have all affixes and synergies aligned. In PvP terms; Flat HP, Flat ED, HP%, Vampiric Touch set, Resist OR Armor, and maybe a sprinkle of critical chance.
As of 20170315, I have played for 2,429 hours on DQ.
I have collected 7 eternal pets in ~40,000 pet drops or a 0.0175% chance for eternal pet drop. This means I experience a pet drop once every 3.6 minutes (2429×60/40,000). During my farming, I have maintained Luck around 1012% and eternal find rate of 650% (current DQ). However, to be fair, my first year, the max luck was 650% and eternal drop rate was low since the ascension perks did not exist and there were only two items (MH & Helmet) that had eternalized set affix.
Important notes below.
There are 27 eternal pets available to drop (9 of 27 pets are elements). This can be expressed as 1/27 or a 3.7% chance to get "Vampiric Touch ~ Lillith", for example.
There are 143 applicable epic affixes. This includes skills and talents from each character. This would be 1/143 or 0.699% chance to obtain Flat ED, for example. There are 5 epic affixes on each eternal pet.
There is approximately 7 in 40,000 odds to obtain an eternal pet or 0.0175% chance (my experience & very similar to other player experience). You can also translate this to mean that you can expect an Eternal Pet once every 5,714 pet drop. Let's say you are three times better than me and most other folks at obtaining eternal pet, then that would be expressed as 21/40000 or a 0.000525 chance of a drop being eternal. If you were 3 times better than myself, you would have an eternal pet drop every 1905 per drop. Yes, this accounts for luck 1012% and eternal find rate of 650%
In each of the 7 eternal pet components below, I am listing the information to show odds of obtaining a perfect eternal pet using factorial combination or hypergeometric distribution mathematics.
Eternal pet chance = 7 in 40,0000
Pet Set chance = 27 (1 in 27 for Vamp Tch)
Epic 1 = 143 (1 in 143 for 20K HP)
Epic 2 = 143 (1 in 143 for 10K ED)
Epic 3 = 143 (1 in 143 for HP%)
Epic 4 = 143 (1 in 143 for Resist)
Epic 5 = 143 (1 in 143 for Crit Chance)
Using factorial combination math we can derive odds below assuming there are 6 affix per pet with 170 possibilities (27 pet set + 143 epic affix).
n = 170
r = 6
Or 3.0663442E10 or 1 in 30,663,442,000 odds without Accounting for low eternal pet drop rate. This means if every pet drop that occurred was eternal, then it would take 972 years to obtain a perfect eternal pet if one dropped every 60 seconds and you played continuously for that amount of time.
Accounting for Eternal Pet odds 1/0.000175 = 5714×3.0663442E10
Or 1.7521091E14 or 1 in 175,210,910,000,000 odds of having a perfect eternal pet. This means if you have an eternal pet drop once every 5,714 Pet drop, then it would take 5,555,992 years to obtain a perfect eternal pet if a pet dropped every 60 seconds and you played continuously for 5M years.
Edit: I relaxed n! and separated 27 pets from n! total)
n=143 & r=5 for epic affix
n=27 & r=1 for pet set
7/40,000 for Eternal pet drop
1 in 7.16323E13 or 1 in 71,632,300,000,000. This means if you have an eternal pet drop once every 5,714 Pet drop, then it would take 2,271,445 years to obtain a perfect eternal pet if a pet dropped every 60 seconds and you played continuously for 2M years.
For an easier explanation, a lottery with 49 available numbers and with six numbers to select gives you odds of winning at 1 in 13,983,816. This can be expressed as;
n = 49
r = 6
One last simple calculation. Let's say you only need 2 of the 6 affixes (maybe Flat ED and Flat HP), then your odds are vastly improved. That means you might get elements set affix and other affixes that are useless (basically you have everything you need in the other 36 affixes but you need these two on pet for double values). This can be expressed as;
n = 143 (not acct for 27 pet set)
r = 2
Or (143/2)×(142/1) or a 1 in 10,153 chance of your eternal pet having both affix. If you are 3x better than me and others obtaining eternal pet then your odds would be 1,905×10,153 or a 1 in 19M chance.
With 143 available epic affixes and with 5 epic slots, it's a very tough task to get a perfect eternal pet so don't be frustrated.
Bottom Line: it is about a one in quadrillion chance to obtain an eternal pet with so many perfect affixes and synergies.