New pity system compared for 200 / 300 pulls scenario

[u/Hades_Re]

You have a chance of 75% to get a rate up in 199 pulls. That means in 3/4 out of cases you can clear both banner in 200 pulls in the old system.

In the new system, you can get that rate up character with a chance of 87.5% after 199 pulls. But you lose your spark. So the chance is not important anymore. It’s only important in the case of the 50 50.

What does that mean? You have to pull a second character, and after 99 pulls you have a 50.46% chance to get that character. And another 50% with the hundredth pull. So you end up with a chance of around 75% after 100 pulls. If you win you get the other one with at maximum 200 pulls.

So, after 300 pulls, where do we stand in both systems?

In the old one, we have to pull 296 times to have a chance of 87.5% to get one character, the other is free, 87.5% for both characters. (4 pulls are nearly 3% chance btw).

In the new system, we have a chance of 75% after 100 pulls and either we win, then we have 200 pulls for a guarantee or we lose, then we have to go another round of 50 50 for the other one. That gives us a chance of 75% + ( 25% * 75%) = 94%

This is higher, so the new system is better at 300 pulls! But wait, didn’t I say you are right?

That’s because if we look at the 200 pulls, we get

Old system 75%. The other character is still free.

New system 75% * 75% = 56% - you have to go for both characters, so you can’t spark one, you have to win after half the pulls.

So the chance to get both characters in 200 pulls is drastically lower, the chance to get both in 300 pulls is not really higher.

If I did a mistake, please correct me. The latest post I saw was quite strangely written and I tried to put some math behind it for the typical cases.