BOTTOM LINE: DO NOT CHASE SPECIFIC CHARACTERS THAT ARE NOT BEING BOOSTED IN A BANNER. WAIT UNTIL THEY ARE BOOSTED.
Update: This is a re-upload of my last post. There's been some significant revelations in my the previous thread from correcting poorly translated details and Mihoyo's chinese publication.
Explanation:
These numbers are the average amount of rolls it would take to get specific characters (starting from pity 0). These are based on simulated pulls and math - not the advertised 'adjusted' 1.6% drop rates for 5* character's since that is very likely false. Keep in mind rolling 697 times for Diluc doesn't mean you just got a Diluc or you're guaranteed to have Diluc, you would have several other 5* characters and weapons by the time you got that many rolls. It just means there's an average of 697 rolls between each Diluc win on that banner, taken over an extremely large sample size.
Math:
I simulated 100,000,000 rolls in VBA and found an average of 1.435% for 5* drops including the guarantee, and 11.83% for 4* including the guarantee. This is lower than their advertised 1.6% for 5*s and 13% for 4* average rates. Similar threads exist going into this in detail already, but here is my code. So with the new average percentage, we can calculate:
100% / 1.435% = 69.68 Average rolls needed for each 5*
100% / 11.83% = 8.45 Average rolls needed for each 4*
Now we can calculate how many rolls needed for a specific 5*, for example to get a specific 5* from the Wanderlust banner you'll first need to roll a 5* (~69.69 rolls) and then you'll have a 50/50 chance of getting either a 5* weapon or 5* character. If you do get a 5* character, you'll have a 1 in 5 chance of getting the character you want (effectively 1 in 10 five star rolls will be the character you want when you include the 50/50 for weapon/character). Finally we can use:
1:10 odds for the character you want * 69.68 avg rolls per 5* = 696.8 average rolls needed for the specific 5* character you want (Wanderlust Invocation)
Odds on the Rate Up banner are a little funky and poorly translated in the details page. But essentially every 5* roll has a 50% chance of being the boosted character, if ANY 5* roll in the boosted banner is NOT the boosted character, then your NEXT 5* roll in the banner is guaranteed to be the boosted character. I'm sure someone better at math can share a formula for these odds, but I just simulated 10,000,000 5* drops with these rules and found an average of 2/3rds of them were Venti and 1/3rd were the other 5* characters.
2:3 (1:1.5) odds for the promo character * 69.68 average rolls for each 5* = 104.5 average rolls needed for the promo character (Promo Banner)
1:3 odds for non-promo 5* character * 1:5 odds for specific non-promo 5* you want * 69.68 average rolls for each 5* = 1045.2 average rolls needed for specific non-boosted 5* you want (Promo Banner)
Real world examples?
A few people have pointed out that when looking at real world data hardly anyone goes the full 90 rolls without getting their 5*. This would suggest there's some kind of scaling increase on the chances of getting a 5* around 70 pity pulls, and also could explain why Mihoyo advertises 1.6% drop rate. I don't have enough data to say for sure, but assuming the real drop rate IS 1.6% instead it would affect the 5* rates like this (4*s would be unchanged):
Permanent banner Promo Baner
Venti N/A 94
Diluc 625 938
Keqing 625 938
Mona 625 938
Qiqi 625 938
Jean 625 938
4 Star characters not in Promo banner:
Amber, Lisa, and Kaeya are not available in the Promo Banner. The averages I listed in the image account for this already, but just something to keep in mind if you're either looking for those drops or are looking to avoid those drops.
Why did you censor P*lls in the thread title?
Moderator bot was deleting my post thinking I was sharing a screenshot of my pull
I thought Venti banner was still better odds for getting Diluc than Permanent banner though?
I did too. We've been mislead by poorly translated details and missing information from Mihoyo. This post goes into more detail on it.
EDIT: However it is worth noting that your FIRST 5* win on the promo banner has equal odds of getting Diluc as a normal 5* win on the Permanent banner. After your first 5* win in the Promo you're better off sticking to the permanent banner if you're only after Diluc (or other non-boosted 5*s)
Gacha Simulator
Here's a third party Chinese site to simulate gacha rolls in this game. I did 100,000 rolls in each banner and found it compares pretty spot on to the averages in my post. I can now finally say I trust this site to be accurate, at least to the extent of what Mihoyo has shared about their rules.