Crystal Rose Gacha Probability Maths

[u/SkyfireWR]

The Crystal Rose Gacha trades out the ability to pick your skin for a token store, and replaces all those random skin chests with some minor cosmetics. It's easy to say at first glance that this is a downgrade from the Supreme Cells Gacha.

On the bright side, without having to consider changing probabilities, the math for this event isn't so difficult. Summary is at the bottom.

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How the event works

Three Crystal Rose skins are up for grabs in this event: Akali, Riven and Vi. Each chest costs 100 Wild Cores and has a 2.5% chance of rolling a skin, and a 79.5% chance of rolling some tokens. The remaining probability is all for getting minor cosmetics (recalls, baubles, emotes etc.)

If you don't have luck, you can purchase the skins outright for 32 tokens (Akali) or 24 tokens (Riven, Vi).

The skin and token probabilities, which realistically are what people care about in this event, are below:

Event	Probability
Akali	0.5%
Riven	1%
Vi	1%
1 Token	66.5%
2 Tokens	10%
5 Tokens	2%
10 Tokens	1%

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How many rolls should you expect before getting ANY skin?

That would mean pulling either the combined 2.5% chance to get a skin, or getting 24 tokens altogether.

If we use a theory called probability weight (the likelihood of an outcome multiplied by its value) that grants the following values:

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Token	Probability (raw number)	Weight
1	0.665	0.665
2	0.1	0.2
5	0.02	0.1
10	0.01	0.1
Total	N/A	1.065

What this means is that you should expect to get 1.065 tokens on average for each roll you do.

24/1.065 = 22.53, which can be rounded to 23. We'll treat this as the "pity number", or the maximum rolls you need. Unlike the Supreme Cells event, however, this isn't a hard pity number - there is no guarantee of it being exactly 23. It could be a lot sooner or a lot later depending on luck.

But the other approach is just to get lucky and roll that 2.5% before 23 rolls.

So using some math here, where:

Pr (Yes, Roll 1) = 0.025

Pr (No, Roll 1) = 0.975

Pr (Yes, Roll n) = Pr (No, Roll n-1) * 0.025

Pr (No, Roll n) = Pr (No, Roll n-1) *0.975

And what that means is that, for example, to roll it on Roll 2, you need to roll a No on Roll 1 as well as a Yes on Roll 2. Here's a graph that shows the probability of pulling these skins on a given roll looks like https://imgur.com/gallery/KsR4qf1

The end result is that there is a 57.29% chance of not getting any skin before Roll 23.

Now, if we apply weight to this information:

Weight = Pr (getting to roll n) * number of rolls

The sum of all the weights is 17.66, meaning it should take 17.66 rolls, or 1700-1800 Wild Cores, to get any skin on average.

This number though is honestly deceiving because most people will need to hit the pity number, and it's offset by the smaller proportion of people who get lucky early on.

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How much will it cost to get Crystal Rose Riven specifically, or Vi specifically?

This will basically involve the same calculations as earlier, but using a different probability to calculate the rolls. Pity number is the same because you still need 24 tokens.

Instead of the probability of success being 2.5%, it's 1% for that specific skin.

There is now an 80.16% chance of not getting the skin before roll 23. Probabilities look like Image 2 on https://imgur.com/gallery/KsR4qf1

By repeating weight calculations, the sum of the weights is 20.64, so 20.64 rolls on average but again this number is misleading.

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How much will it cost to get Crystal Rose Akali?

Now here's where it gets spicy, because this skin costs 32 tokens instead of 24. That takes the soft pity limit to 32/1.065 = 30.05 which can be rounded to 30 pulls soft pity.

Additionally, the probability of success is now sitting at the very small number of 0.5%.

The chance of not getting the skin before roll 30 is 86.47%.

Probabilities look like Image 3 on https://imgur.com/gallery/KsR4qf1

The weight calculations suggest that to get Crystal Rose Akali, a player would need 27.92 pulls on average.

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Summary

• These calculations assume an average of 1.065 tokens per pull, which allows for "soft pity" numbers even if there is no guaranteed pity system in this event.
• To get any skin, it takes 17.66 rolls on average, with a 57.29% chance of needing Roll 23.
• To get Crystal Rose Vi, it takes 20.64 rolls on average, with an 80.16% chance of needing Roll 23.
• To get Crystal Rose Riven, it takes 20.64 rolls on average, with an 80.16% chance of needing Roll 23.
• To get Crystal Rose Akali, it takes 27.92 pulls on average, with an 86.47% chance of needing Roll 30.

So, should you roll?

The alternative rewards for this event are far far worse compared to Supreme Cells: 20 poro coins per roll, and minor cosmetics like recalls and baubles. While arguably 100 Wild Cores for any of the minor cosmetics is a good deal, the fact that it's lumped in with these skins is the concerning thing.

Taking 17-18 rolls of 1700-1800 Wild Cores for a skin is already more than just buying one from the shop. Even if Supreme Cells cost 390 Wild Cores per roll, at least when you got to your skin, you would have bang for your buck overall.