I'm trying to come up with an easy to digest table and have compiled the necessary data. I have been googling how to calculate multiple probabilities but none of what I could found can seem to replicate Nazta's odds of pulling a regina. It just might be that my formulas are correct but my probabilities aren't.

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TL;DR:

• If you'll spend 10k lapis, summon on the 5k 5% 5* Banner Twice
• If you'll spend around 25k~30k lapis, do step-ups
• If you'll spend around 40k lapis, do 1 full lap of step-up, 1st step of the 2nd lap for that 5* guaranteed 30% featured unit, then spend the remaining 10k on the 5% 5* Banner
• Bonus: If you don't care about the on banner units, two summons of Seasonal Celebration is better than 5k 5% Banner as it guarantees 2 rainbows

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I made this thread to help myself and anyone in deciding whether which banner is best depending on their current lapis hoard and what the true value is with each kind of summon banners Gumi has been giving us. Kind of like a bundle review but for summon banners instead.

Before I proceed further, I'd like to give huge thanks to u/bobusisalive for linking his post to me which allowed me to figure out what's wrong with my calculations and probabilities. It has taught me everything I needed to know to be able to replicate u/Nazta's odds of pulling a regina and populate the table below.

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Now I'd like to point you specifically towards the ones that cost around 25k~30k. At first glance, 5% rainbow rate sounds wonderful for pulling a 5* on banner unit. But in actuality, it doesn't even come close to the step-ups. 30k lapis on the 5% rainbow only gives an 83% chance of getting 1 on banner unit, whereas spending the same amount on the step-ups already gives us approx 75% chance of getting a 2nd 5* on banner unit.

5% rainbow rate banners aren't necessarily bad though because if you're only willing to spend 10k lapis then this banner is 100% a better choice.

Running two steps on a generally 5k per step-up only nets 28.38%.

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So if you find yourself willing to spend 40k lapis, you're best approach is to run through 1 lap 1st step on the 2nd lap and then switch banners and pull for the 2-time 5% 5* banners.

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Assumptions derived using featured / off-banner ratios:

In a 3% rainbow rate:

On a 3* minimum summon, the chances are - 1% 5* banner unit, 2% 5* off-banner unit, 19% 4*, and 78% 3*

On a 4* minimum summon, the chances are - 3.75% 5* banner unit, 1.25% 5* off-banner unit, and 95% 4*

What this tells me is that on 3* min, the 5* banner unit rate is 1/3rd of rainbow rate, whereas on the 4* min, the 5* banner rate is 3/4ths of the rainbow rate

As such, for the 5% rainbow rate, I'll be using 1.6667% (1/3rd of 5%) for the 3* minimum and 5.25% (3/4ths of 7%) for the 4* minimum

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Many thanks to any and all assistance and apologies for this math heavy thread.

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Initial Notes:

Using Probability of a Series of Events

https://www.calculator.net/probability-calculator.html

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Taken from the GL - Maint, Story Event: Regina, Item World, 5% Rainbow Banner - 6/6 :

Note: Odds assuming Rainbow rates of 5% (7% 1) with commonly used featured / off-banner ratio.

• 1x = 44.3% Rainbow (19.9% Regina)
• 2x = 68.9% Rainbow (35.8% Regina [w/ 20%: 48.6%])
• 3x = 82.7% Rainbow (48.6% Regina)
• 4x = 90.3% Rainbow (58.8% Regina)
• 5x = 94.6% Rainbow (67.0% Regina)
• 6x = 97.0% Rainbow (73.6% Regina [w/ 2x 20%: 83.1%])

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The nearest I could find was P(a) P(b) P(c) minus P (neither (a), nor (b), nor (c) occurring):

P(a) = Min 3* (3.75% 5* Banner Unit)

P(b) = Min 4* (5% 5* Banner Unit)

P(c) = 5* Feature Unit Summon Ticket (20% 5* Banner Unit)

=((1-(1-0.0375)^20) (1-(1-0.05)^2) (1-(1-0.2)^1))-(((1-0.0375)^20)*((1-0.05)^2)*((1-0.2)^1))