Nexus Crystal Synthetic Drop Odds

The drop odds for the Nexus PHC are the same, per selection, as the normal PHC - 3% 4*, 20% 3*, 77% 2*. But what are the odds of getting each tier from the Nexus, given the triple choice?

If we assume that a player will always select the highest star rating champion (or from the highest if there are multiple), then we can simplify this a bit. The odds of pulling three 2* champs (and thus forcing you to pick a 2*) is (0.77)^3 ~ 0.4565, or about 45.65%. The odds of pulling at least one 4* champ is a bit more complicated to calculate, but it comes out to be ~ 0.08733 or about 8.73%. That means the odds of getting at least one 3* but no 4* is 100-45.65-8.73 = 45.62%.

So basically, the odds of "effectively pulling" the different tiers in the Nexus PHC are:

45.65% 2*
45.62% 3*
8.73% 4*

Given that it costs 50% more than the base PHC, that's a substantial increase in value.

Just for giggles, what would the synthetic drop odds be for a hypothetical Cavalier crystal? Glad you asked. The standard Cav has drop odds of: 1%/11%/38%/50%. So the odds of getting stuck with three 3* champs in a Nexus Cav would be (0.5)^3 = 0.125, or 12.5%. The odds of getting at least one 4* but no 5* or 6* champs would be about 55.65%. The odds of getting at least one 5* but no 6* would be about 28.88%. And the odds of getting at least one 6* champ would be about 2.97%.

So the synthetic drop odds of getting at least a certain tier would be about 3%/28.9%/55.7%/12.5% (rounded to one decimal place). The odds of getting a 5* or 6* champ would be almost one in three.

Comments

Hendross

Spot-on binomial probability distribution. 1 caveat is if the trials are with or without substitution, which is unknown but would not be a material change to the above posted probabilities. I can't wait til someone rolls triple Groots and the Kabam servers catch on fire (again).

DNA3000

Spot-on binomial probability distribution. 1 caveat is if the trials are with or without substitution, which is unknown but would not be a material change to the above posted probabilities. I can't wait til someone rolls triple Groots and the Kabam servers catch on fire (again).

My guess is that the drops are independent, mainly because it is actually harder to shuffle cards than roll dice, as we say. And as you point out, there's enough different champs to make the difference too small to really notice.

Apollo107

Tried one and got 2*s for all

Lormif

Spot-on binomial probability distribution. 1 caveat is if the trials are with or without substitution, which is unknown but would not be a material change to the above posted probabilities. I can't wait til someone rolls triple Groots and the Kabam servers catch on fire (again).
My guess is that the drops are independent, mainly because it is actually harder to shuffle cards than roll dice, as we say. And as you point out, there's enough different champs to make the difference too small to really notice.

You can do it pretty easily, just remove that champ from the list for the 2nd and the 2 champs from the roll for the 3rd. I fear it was not done that way but we will see if the complains come in.

SummonerNR

Has anyone actually gotten a choice of champs that were of DIFFERENT Star Ranks ??

I would assume how this crystal works is that it Random picks which Star Rank you get, and THEN would randomly choose which of the 3 champs to offer you “of that already determined Star Level”.

So that Drop Rates are still identical to the base Drop Rates.

Lormif

Has anyone actually gotten a choice of champs that were of DIFFERENT Star Ranks ??

I would assume how this crystal works is that it Random picks which Star Rank you get, and THEN would randomly choose which of the 3 champs to offer you “of that already determined Star Level”.

So that Drop Rates are still identical to the base Drop Rates.

seatin did, he got a 4* and 2 2*s

DNA3000

Spot-on binomial probability distribution. 1 caveat is if the trials are with or without substitution, which is unknown but would not be a material change to the above posted probabilities. I can't wait til someone rolls triple Groots and the Kabam servers catch on fire (again).
My guess is that the drops are independent, mainly because it is actually harder to shuffle cards than roll dice, as we say. And as you point out, there's enough different champs to make the difference too small to really notice.
You can do it pretty easily, just remove that champ from the list for the 2nd and the 2 champs from the roll for the 3rd. I fear it was not done that way but we will see if the complains come in.

I'm not sure what you mean by "easily." If you're just writing your own python program or something, sure, it is easy to complete this CompSci101 homework assignment. But in a game like this, rewards are almost always implemented using reward tables not lists, and there's no obvious way to "remove" a reward from the reward table so it isn't selected again. That would require specific reward engine code to be added, which means the difference in effort between independent drops and dependent drops is about ten thousand to one. Independent drops already exist and take zero extra effort. Dependent drops take actual C code inserted into the engine, which requires programmer time, design docs, resource impact review, producer approval, QA testing, toolchain additions, new reward tables, new data merge tools, schema changes, and live server merges.

SummonerNR

Has anyone actually gotten a choice of champs that were of DIFFERENT Star Ranks ??

I would assume how this crystal works is that it Random picks which Star Rank you get, and THEN would randomly choose which of the 3 champs to offer you “of that already determined Star Level”.

So that Drop Rates are still identical to the base Drop Rates.
seatin did, he got a 4* and 2 2*s

Wow, that’s rough then, that’s really not a “choice” if you have to choose between a 4* and a 2*.

Guess it’s better at least from the point that it’s really better odds then for at least some type of 4*. But still...

DNA3000

I would assume how this crystal works is that it Random picks which Star Rank you get, and THEN would randomly choose which of the 3 champs to offer you “of that already determined Star Level”.

I would assume the most straight forward way to implement the nexus PHC is for the game to open three perfectly normal PHCs in the background, show you the results, and let you choose one. What you're describing is basically how a single normal PHC works: a random roll happens and the PHC picks from one of three reward subtables: 2*, 3*, and 4*. And then it rolls again to see which champ on that particular table you get.

DNA3000

Has anyone actually gotten a choice of champs that were of DIFFERENT Star Ranks ??

I would assume how this crystal works is that it Random picks which Star Rank you get, and THEN would randomly choose which of the 3 champs to offer you “of that already determined Star Level”.

So that Drop Rates are still identical to the base Drop Rates.
seatin did, he got a 4* and 2 2*s
Wow, that’s rough then, that’s really not a “choice” if you have to choose between a 4* and a 2*.

Guess it’s better at least from the point that it’s really better odds then for at least some type of 4*. But still...

Suppose the Nexus crystal worked this way. You roll a PHC. If you like it you keep it, if you don't like it, you can reroll up to two times. That's a reasonable "choice."

The Nexus is better. It rerolls for you twice and lets you pick any of the results: the original, or any of the two "rerolls." That's better than being able to reroll, because you can pick an earlier option if the later ones are not as good.

Lormif

Spot-on binomial probability distribution. 1 caveat is if the trials are with or without substitution, which is unknown but would not be a material change to the above posted probabilities. I can't wait til someone rolls triple Groots and the Kabam servers catch on fire (again).
My guess is that the drops are independent, mainly because it is actually harder to shuffle cards than roll dice, as we say. And as you point out, there's enough different champs to make the difference too small to really notice.
You can do it pretty easily, just remove that champ from the list for the 2nd and the 2 champs from the roll for the 3rd. I fear it was not done that way but we will see if the complains come in.
I'm not sure what you mean by "easily." If you're just writing your own python program or something, sure, it is easy to complete this CompSci101 homework assignment. But in a game like this, rewards are almost always implemented using reward tables not lists, and there's no obvious way to "remove" a reward from the reward table so it isn't selected again. That would require specific reward engine code to be added, which means the difference in effort between independent drops and dependent drops is about ten thousand to one. Independent drops already exist and take zero extra effort. Dependent drops take actual C code inserted into the engine, which requires programmer time, design docs, resource impact review, producer approval, QA testing, toolchain additions, new reward tables, new data merge tools, schema changes, and live server merges.

if it is a list or a reward table there is still a % chance associated with each prize and its unique id, or its value. You can make the subsequent rolls drop by the % chance to roll the previous prize then shift the rewards upwards at or above that prize level. The second one becomes a little trickery but still possible.

StevieManWonder

Thanks for the info @DNA3000

DNA3000

Spot-on binomial probability distribution. 1 caveat is if the trials are with or without substitution, which is unknown but would not be a material change to the above posted probabilities. I can't wait til someone rolls triple Groots and the Kabam servers catch on fire (again).
My guess is that the drops are independent, mainly because it is actually harder to shuffle cards than roll dice, as we say. And as you point out, there's enough different champs to make the difference too small to really notice.
You can do it pretty easily, just remove that champ from the list for the 2nd and the 2 champs from the roll for the 3rd. I fear it was not done that way but we will see if the complains come in.
I'm not sure what you mean by "easily." If you're just writing your own python program or something, sure, it is easy to complete this CompSci101 homework assignment. But in a game like this, rewards are almost always implemented using reward tables not lists, and there's no obvious way to "remove" a reward from the reward table so it isn't selected again. That would require specific reward engine code to be added, which means the difference in effort between independent drops and dependent drops is about ten thousand to one. Independent drops already exist and take zero extra effort. Dependent drops take actual C code inserted into the engine, which requires programmer time, design docs, resource impact review, producer approval, QA testing, toolchain additions, new reward tables, new data merge tools, schema changes, and live server merges.
if it is a list or a reward table there is still a % chance associated with each prize and its unique id, or its value. You can make the subsequent rolls drop by the % chance to roll the previous prize then shift the rewards upwards at or above that prize level. The second one becomes a little trickery but still possible.

I'm not sure who you're trying to convince that this is possible. I said it was possible. But logistically, it is so much more effort for exactly zero benefit. Doing this is not a requirement, so you'd be going way out of your way to accomplish the unremarkable.

Also, the way you're describing it is not the way I would do it. You end up with all sorts of potential fencepost and round off errors, and this requires a lot of extra state. The proper way to do this is probably to use collision bitmaps. If you're crazy, you can also do offset rolls, but that just seems to be asking for trouble unless you were under some sort of efficiency gun.

And this bears repeating: the time to write the code fragment to make this work is 0.1% of the total resource burn doing this would eat up in any game development group, or for that matter probably most anything development groups.

unknown

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Aleor

The odds of pulling at least one 4* champ is a bit more complicated to calculate

Why? It's almost as simple: 1 - (1-0.03)^3
Like the probability of any star except 4 happening 3 times

DNA3000

The odds of pulling at least one 4* champ is a bit more complicated to calculate
Why? It's almost as simple: 1 - (1-0.03)^3
Like the probability of any star except 4 happening 3 times

That's a bit more complicated to explain, and I didn't want to explain all the calculations here. From my experience, most people who understand anything at all about probability understand the odds of three things happening in a row is n^3, so I mention that. The calculation you're showing is basically the calculation I used for every other number, which is the odds of a 4* not happening three times in a row, not happening. Not everyone understands that one.

DNA3000

@DNA3000 could you calculate the odds for if they did it for cavalier crystals

I did that in the second half of the original post.