These are the real 5stars of Genshin Impact, so let's try to put it in to numbers and understanding the artifact hell.
DISCLAIMER: THIS IS A BIOLOGY MAJOR ATTEMPTING TO USE MATH, RESULTS MAY BE INACCURATE!
I want to base this off 20 resin runs rather than a condensed resin as that seems much more complicated. So to begin with within each artifact domain there are two sets of 5 star artifacts, each obviously including 5 different pieces: Circlet, Plume, Goblet, Flower, Sands. Now without including the minimal percentage of getting two artifacts instead of one with 20 resin, we have ourselves a 1/10th or 10% chance of acquiring the piece we want. This is of course assuming that the drop rate of each piece is equal. Now let's take this 10% and add other levels of random probability.
Circlet of Logos:
Possible main stats: HP%,ATK%,DEF%,CRITR%,CRITD%,HEALING%,EM- 1/7th or 14% chance of getting the exact main stat you need assuming the rates are equal among all main stats.
Possible sub stats: ATK,DEF,HP,ATK%,DEF%,HP%,EM,ER%,CRITR%,CRITD%- 1/10th or 10% for a first substat you want, 1/9th or 11% for a second substat you want, 1/8th or 12% for a third substat you want, and 1/7th or 14% for the final substat you want. This gives us a whopping .018% chance of rolling an artifact with four desirable substats. These assumptions could vary on the mainstat as getting anything but healing% would change the odds by -1, so you'd begin with 1/9th at the first substat, but this is just to keep it simple.
Total Circlet Probability: 10% chance drop x 14% chance desirable mainstat x .018% chance of perfect substats= .000257% chance of acquiring the perfect circlet per 20 resin.
Sands of Eon:
Possible main stats: HP%,DEF%,ATK%,ER%,EM- 1/5th or 20% chance of acquiring the desirable mainstat assuming all rates are equal.
Possible sub stats: Refer to circlet as this calculation is the same among sands, circlet, and goblet.
Total Sands Probability: 10% x 20% x .018%= .00037% of acquiring the perfect sands per 20 resin.
Goblet of Eonothem:
Possible main stats: ATK%,DEF%,HP%,EM,PYRO%,CRYO%,ANEMO%,HYDRO%,ELECTRO%,GEO%,PHYSICAL%- 1/11th or 9% chance of getting the desirable mainstat assuming rates are equal.
Possible sub stats: refer to circlet
Total Goblet Probability: 10% x 9% x .018%= .000166% chance of acquiring the perfect goblet per 20 resin.
Plume of Death:
Possible main stat: exclusively flat ATK, no calculation or rng here.
Possible sub stats: Now because the plume will always roll flat atk, we will always be working with 9 possible sub stats CRITR%,CRITD%, HP%,HP,ATK%,DEF,DEF%,ER%,EM. 1/9th or 11% of desirable first stat, 1/8th or 12% of desirable second stat, 1/7th or 14% of desirable third stat, and 1/6th or 16.6% chance of desirable final stat. This gives us a .031% chance of getting four desirable substats
Total Plume Probability: 10% x .031%= .0031% of acquiring the perfect plume per 20 resin.
Flower of Life:
Possible main stat: exclusively flat HP
Possible substats: Refer to plume, only difference would be elimination of flat hp rather than atk but calculations remain the same.
Total Flower Calculation: 10% x .031%= .0031% of acquiring the perfect flower per 20 resin.
Future Suggestions/Outlook:
When looking at the randomness each player must face when farming for artifacts, it becomes apparent that resin and the accumulation of it is possibly not the issue within this game. It is rather the fact that the area where a huge chunk of your character's stats comes from is filled with so much probability. Within Gacha games there is of course always a major grind and very long processes involved with upgrading your characters, but within that grind there is always a starting line and a finish line. You're constantly moving the whole time as you consume the games version of resin even if ever so slowly you will reach that finish line one day. This isn't the case here as you can go weeks spending all your resin into artifact domains and make 0 progress. It's discouraging and bad game design as you eventually feel you're progression locked no matter how much you play.
How can we fix this? The least favorable but possible solution is increasing the number of 5 star artifact drops per run, this won't fix the randomness but will increase the likelihood of probability being in your favor one day. Another great solution is to simply add a method that always you to reroll the stats on an existing artifact, or a method of being able to spend resources and pick a specific stat that you want. They could make this costly and time consuming to prevent players from just getting everything they want quickly, but the fact that it exists is enough as it gives hope to unlucky players that they will one day maximize their favorite characters! Of course I'm open to other suggestions as this is a community discussion.
