if u open 10 crystals at a 4.5% chance of a 6* u have a 36.9% chance of getting 1 6* if u open 20 at a 3% chance of a 6* u have a 45% chance to get 1 6*

we r actually better off now than before if the number of shards is the same

Not exactly. Chances are independent, thus multiplicative not additive.

Let's say we would get 20,000 shards.

You could get 5 crystals with a 4.5% chance of a 6* (or 95.5% of something else). The chance of not getting a 6* is 95.5%^5 = 0.794%, or 20.6% chance of a 6*.

You could get 10 crystals with a 3% chance of a 6* (or 97% of something else). The chance of not getting a 6* is 97%^10 = 0.737%, or 26.3% chance of a 6*. A slight improvement.

That is, we will be getting the same amount of shards.

if u open 10 crystals at a 4.5% chance of a 6* u have a 36.9% chance of getting 1 6* if u open 20 at a 3% chance of a 6* u have a 45% chance to get 1 6*

we r actually better off now than before if the number of shards is the same

Not exactly. Chances are independent, thus multiplicative not additive.

Let's say we would get 20,000 shards.

You could get 5 crystals with a 4.5% chance of a 6* (or 95.5% of something else). The chance of not getting a 6* is 95.5%^5 = 0.794%, or 20.6% chance of a 6*.

You could get 10 crystals with a 3% chance of a 6* (or 97% of something else). The chance of not getting a 6* is 97%^10 = 0.737%, or 26.3% chance of a 6*. A slight improvement.

That is, we will be getting the same amount of shards.

We're you qouting someone else? You said "not exactly" and then proceed to explain the exact same thing with exact same conclusion but with different number of crystals?

if u open 10 crystals at a 4.5% chance of a 6* u have a 36.9% chance of getting 1 6* if u open 20 at a 3% chance of a 6* u have a 45% chance to get 1 6*

we r actually better off now than before if the number of shards is the same

Not exactly. Chances are independent, thus multiplicative not additive.

Let's say we would get 20,000 shards.

You could get 5 crystals with a 4.5% chance of a 6* (or 95.5% of something else). The chance of not getting a 6* is 95.5%^5 = 0.794%, or 20.6% chance of a 6*.

You could get 10 crystals with a 3% chance of a 6* (or 97% of something else). The chance of not getting a 6* is 97%^10 = 0.737%, or 26.3% chance of a 6*. A slight improvement.

That is, we will be getting the same amount of shards.

We're you qouting someone else? You said "not exactly" and then proceed to explain the exact same thing with exact same conclusion but with different number of crystals?

The conclusion is highly dependent on the amount of crystals. I don't think we ever got 10 cav crystals. Although 5 might be underestimating.

You could get 7 crystals with a 4.5% chance of a 6* (or 95.5% of something else). The chance of not getting a 6* is 95.5%^7 = 72.4%, or 27.6% chance of a 6*.

You could get 14 crystals with a 3% chance of a 6* (or 97% of something else). The chance of not getting a 6* is 97%^14 = 65.3%, or 34.7% chance of a 6*. A slight more improvement.

Always check the drop rates lol. Kabam have a very loose relationship with integrity

Is it unethical or wrong of them to just provide a different reward? Cav crystals is a title that applies to both featured and regular cavs. Unsure why you are jumping to the assumption that Kabam tries to trick us with somehow inferior rewards even though they clearly show us the drop rates on the crystals so it shouldn't actually go unnoticed by anyone.

Why do people assume we get more crystals just because the price is 2k shards? If we get 29 potential crystals this month if we buy all the instore offers, how many did we get last month? I am by no means saying it is better or worse, just questioning the logic of getting more crystals just because they dropped the price. They control the flow of how many shards we can get as well. Might balance out in the end

We can get 20 in the Store alone, plus the Calendar.

Another L from Kabam, I really fail to understand them.

Apparently not the only thing you fail to understand if you read through the comments.

I understand the probability but the thing is Kabam should put changed rates in their notes.

Also this time the shards are optional we are not getting all shards so it's not 1:2 ratio at all.

These are *free* crystals. They don't "have" to put into notes.

So you choose to forgo the crystals and want to complain about reduced drop rates for what reason?

And you get 15 from the calender so there is no situation that you're worse off here

Why does everybody say anything we get from completing some content is free.

Why I don't understand, it's not free I worked for it and spent my time on it .

Free is something for which I did not have to do anything.

Did you totally ignore, the emphasis on the word free? I put it there to distinguish between free rewards as in those that you don't spend units or money on

It's really not that complicated to understand. One chance at 4.5% is not better than 2 chances at 3. If the objective is to acquire a 6*, twice the chances at 1.5% less is still better. The Drop Rate is 3, but you're rolling it twice.

I mean, in a practical sense, you could get one 6* in the 4.5, none in it, two in the 3%, any combination. Either way, with a difference of 1.5%, two rolls are better in my opinion. Now, if that difference were greater, it may be a different story.

I think there are 22 crystals available throughout the event but I could be wrong. So if you open up 22 there is a 50ish percent chance that you pull a six star.

if u open 10 crystals at a 4.5% chance of a 6* u have a 36.9% chance of getting 1 6* if u open 20 at a 3% chance of a 6* u have a 45% chance to get 1 6*

we r actually better off now than before if the number of shards is the same

Not exactly. Chances are independent, thus multiplicative not additive.

Let's say we would get 20,000 shards.

You could get 5 crystals with a 4.5% chance of a 6* (or 95.5% of something else). The chance of not getting a 6* is 95.5%^5 = 0.794%, or 20.6% chance of a 6*.

You could get 10 crystals with a 3% chance of a 6* (or 97% of something else). The chance of not getting a 6* is 97%^10 = 0.737%, or 26.3% chance of a 6*. A slight improvement.

That is, we will be getting the same amount of shards.

We're you qouting someone else? You said "not exactly" and then proceed to explain the exact same thing with exact same conclusion but with different number of crystals?

The conclusion is highly dependent on the amount of crystals. I don't think we ever got 10 cav crystals. Although 5 might be underestimating.

You could get 7 crystals with a 4.5% chance of a 6* (or 95.5% of something else). The chance of not getting a 6* is 95.5%^7 = 72.4%, or 27.6% chance of a 6*.

You could get 14 crystals with a 3% chance of a 6* (or 97% of something else). The chance of not getting a 6* is 97%^14 = 65.3%, or 34.7% chance of a 6*. A slight more improvement.

Overall, the more crystals, the better for us.

Although I'm quoting one poster, this is really a reply to the entire thread of thought.

Once upon a time, the math to calculate the chance for something to happen or not happen was not common knowledge here on the forums. I'm not sure I would call it common knowledge now, but it is much more commonly discussed. The problem is that it is now being applied universally to all statistical situations whether it actually applies or not.

When we're talking about a situation where a player is attempting to achieve something, one very specific something, that sort of math can calculate the chance of binary success: yes or no. Do I land the featured champ or not? Do I get a 6* champion or not? That sort of thing.

But when it comes to expected value that math doesn't apply, because players do not generally value things based on binary win/loss criteria. One 6* and two 6* champs are both "winning" in such calculations, but they obviously aren't worth the same amount.

If we're judging the value of a 4.5% 6* crystal or a 3% 6* crystal that costs half as much, ironically the simple calculation is the correct one. One 4.5% 6* crystal, if all we care about is 6* champs, is worth 1.5 3% crystals. If the 3% crystals cost half as much, then they have more value (this presumes all other things being equal, which may or may not be true depending on the availability of the shards to form them). To put it another way, the expensive crystal offers 50% more 6* champs for 100% more cost. It is not as good of a value.

If I had X number of shards and I had a choice: buy 4.5% crystals at 4000 shards per crystal or buy 3% crystals at 2000 shards per crystal, then assuming I have some multiple of 4k shards (so I'm buying a whole number of each regardless) I'm buying the 3% crystals. They offer the best return.

I think maybe the issue here is that a lot of what we are discussing is based on assumption.

There are only 3 facts we know for sure right now

1) The drop rate have been decreased for this month event (we don't know if this is permanent or if it would change back im the next month event)

2) We are getting more Event crystals this month, this is with a little assumption, as we won't know for sure until the event end. (However we also don't know if Kabam will give similar amount going forward, if next Event it drop to 7 Event crystals, in which it would be worst than before).

3) Kabam have not officially announced these changes or plan going forward. So it kind of leave players confused and fearful.

As for my opinion, with so much previous issues of crystals coming out with wrong drop rates (in the pass few months), is it really worth for Kabam to make another set of different crystals rate?

Feel like it just adding more work and info into a systems that is already overloaded data. Feel like extra work for the already overworked development team.

Either way I think it would be helpful to get an official response from the admin team

Another L from Kabam, I really fail to understand them.

Apparently not the only thing you fail to understand if you read through the comments.

I understand the probability but the thing is Kabam should put changed rates in their notes.

Also this time the shards are optional we are not getting all shards so it's not 1:2 ratio at all.

These are *free* crystals. They don't "have" to put into notes.

So you choose to forgo the crystals and want to complain about reduced drop rates for what reason?

And you get 15 from the calender so there is no situation that you're worse off here

Why does everybody say anything we get from completing some content is free.

Why I don't understand, it's not free I worked for it and spent my time on it .

Free is something for which I did not have to do anything.

Game developer gives you a free 6-Star crystal. Unless you login (work and time), you don't get it. So it is a free crystal or not.

Your govt send you a voucher to redeem cash at their office. It is free. Do you need to drive a car, walk there or do something?

This is on a whole new level..lol. You are equating cash to a 6 star crystal? Tell me if you can exchange that crystal for food? Clothes? Maybe pay your rent? Forget all that..can you exchange that 6 star crystal with a couple of t5b? I don’t think so. That crystal has absolutely zero value in the real world. So poor analogy. Besides the person you are responding to is not talking about login or calendar crystal but the ones you get for completing content. You do work for those. They are not free.

I mean, in a practical sense, you could get one 6* in the 4.5, none in it, two in the 3%, any combination. Either way, with a difference of 1.5%, two rolls are better in my opinion. Now, if that difference were greater, it may be a different story.

But you’re assuming that the only two choices were

A) what we have, and same number of shards but betters odds at double the shards needed per crystal.

You’re right about the statistics, but it’s a disappointing “oversight” that they failed to mention the crystals reverting to the old rates even though they announced the 2000 shards per crystal.

I’d rather have twice as many pulls with a 33% lower drop rate. 4 with a 2% drop rate? Even better. 8 with a 1.33 drop rate? Butter my pancakes and call me Billy- it’s a deal.

So....one chance at 4.5 is greater than 2 chances at 3? Curious about that math.

Just curious if you will take a crystal with 100% chance of a six star or 2 crystals with 50% chance of a six star?

I don't believe anyone would take that chance unless they were fond of gambling, but we're not dealing with absolutes here. We're dealing with RNG. One may be 1.5% higher, but 2 spins is twice the chance.

I'm bored so I'll dump some quick maths before DNA can

You roll 3% once, it becomes 3/100 to get a six star. You roll 3% twice and you end up with a few different paths.

If didn't pull a six star from either crystal, (97/100)(97/100) would be equal to around 9,409/10000. We can shorten this down to 94.09% of you not getting a six star from either crystal.

If you pulled a six star from the first one but not the second one, (3/100)(97/100) would be equal to around 291/10000. You can shorten this down to 2.91%.

If you didn't pull a six star from the first but did on the second, (97/100)(3/100). Once again you would end up with 2.91%.

Now if you're the lucky **** that managed to get six stars in both crystals in a row,(3/100)(3/100) nets you 9/10000 odds. This is roughly equal to 0.09%.

Now if you add the events that involve you getting six stars. You have a 5.91% chance of pulling at least one 6* from two crystals.

Assuming that the shard spread did not decrease from say... last months events.

You have 1 4.5% chance against 1 5.91% chance once you even it out. But that's only if the shard spread didn't change.

If anyone has any idea how many crystals you could get without using the top ups in the stores from last month and this month, then we can make a reliable statement. But right now as I see it, these crystals are technically better.

If we're judging the value of a 4.5% 6* crystal or a 3% 6* crystal that costs half as much, ironically the simple calculation is the correct one. One 4.5% 6* crystal, if all we care about is 6* champs, is worth 1.5 3% crystals. If the 3% crystals cost half as much, then they have more value (this presumes all other things being equal, which may or may not be true depending on the availability of the shards to form them). To put it another way, the expensive crystal offers 50% more 6* champs for 100% more cost. It is not as good of a value.

I’m not sure of what you mean with this. Reading it quickly it gave the me impression that the correct calculation of how many 3% crystals is worth a 4.5% crystal if we want one or more 6* comes simply from the probability ratios p1/p2. Having read some of your posts it's likely I interpreted you incorrectly and you were referring to the cost/return part at the end of the quote (which I agree with). But to avoid any confusions, let’s expand on it.

If you got pass the typos due to writing fast, the probability breakeven by trial numbers I calculated in the previous page for getting at least 1 6* from the two different drop rates was:

N2 = N1 * log(1-p1)/log(1-p2) [1]

Which happens to be close to 1.5 and is very simple math, but it is not coming from p1/p2. Now let’s use the ratio p1/p2 = 1.5 as a condition to see how the expression [1] behaves:

N2/N1 = log(1-1.5p2)/log(1-p2)

Under p1 = 1.5*p2 boundary, N2/N1 has this shape :

N2/N1 match p1/p2 ratio = 1.5 at very low p2 values. From there, the logarithmic functions take over.

As an example

let’s say we pick p2 = 0.5 and p1 = 0.75 (yes, too generous of a crystal, I know). The ratio p1/p2 is still 1.5, but we won’t match the probability of getting at least a 6* until the crystal ratio is 2, or N2 = 2 N1. Going farther means you would need to open more than twice the number of crystals with p2 to even out the chances of opening at least 1 6* from the ones with p1.

Let’s find an independent source to test it: I picked the first link returned googling chance calculator. If we try and match the number of trials to get equal probability of having one or more for the special case p2 = 0.5 and p1 = 0.75, we get for N1=2:

And we match probabilities when the number of trials satisfy N2 = 2 N1.

To summarize my point, while I understand what you say about focusing on the expected value to estimate the crystal equivalence (E[X1] = E[X2]; n1p1 = n2p2 -> n2/n1 = p1/p2), this estimation is informative only if the law of the large numbers apply. That is, if each one of us get to open hundreds of crystals.

It will also work well to control the global resources inflow when changing the drop rates, since it will measure all crystals opened and not a specific account.

But when you have very low sample sizes (like 10-20), the averages you will get will vary wildly from E[X] = np. Also, given how many times I've seen people upset because they have opened, say, 5 crystals and not gotten 1 of something with a 20% chance, in my opinion using the expected value can lead to confusion.

That's why I personally find more informative to compare the probabilities of getting 'at least' 1 6*. And yes, even though getting 1 or 2 or 3 is not the same, considering how low the chances are and the small amount of crystals we get, I would consider 'at least 1' a success.

That being said, the outputs of both approaches are indistinguishable due to the probabilities being very low, so all this text I posted (correct or not) is probably irrelevant and a massive waste of time to anyone reading. Sorry for that!

I love the great Maths debate (even though I don't get it).

The only thing I think I got is that, whether it is better or worst to have lower drop rate with more crystals.

But the key variable that we don't know is will we continue to get more (double the amount) crystals than previous event when the drop rate was higher?

We get more this month, but what is the plan going forward. This where is would be help to get an official response from Kabam

To summarize my point, while I understand what you say about focusing on the expected value to estimate the crystal equivalence (E[X1] = E[X2]; n1p1 = n2p2 -> n2/n1 = p1/p2), this estimation is informative only if the law of the large numbers apply. That is, if each one of us get to open hundreds of crystals.

Actually, this is just the definition of "expected value."

You're heading away from the realm of statistics and into the realm of psychology. When we talk about expected value, we mean objective expected value. The value that we can assign mathematically given only that the factors involved are numerically quantified ones. One 6* is twice as valuable as two. If you leave this objective viewpoint you can make anything worth anything, but if you decide to invent an alternate definition of expected value then you'll leave the realm where any other statements anyone makes about value may not apply to you anymore. For example, if you expect the game design to honor that definition of value, that's never going to happen.

You might ask why this definition is the one everyone agrees to, and not some other: in other words, is the definition arbitrary, and any other is equally good. In fact, this definition of objective expected value is the only one that tends to make sense when judging games of chance for a very wide range of reasons. Game theory tends to become nonsensical if you break the definition of expected value. But that's a very extensive discussion.

You're certainly free to say "well, for me all I want is at least one 6*. I don't care if I get one or ten, I want at least one" and then calculate which crystal opportunity satisfies that constraint. Just recognize this isn't how value is generally defined, and thus all situations you're ever likely to find yourself in won't be using it and will likely be highly inconsistent with it.

To summarize, if Kabam makes a crystal with a 20% chance to drop a 6* that costs 4000 shards, and then later makes a crystal with a 10% chance to drop a 6* that costs 2000 shards, from their point of view those two crystals will have equal value (given the same availability of shards) because that's the generally accepted definition of value. Any counterargument that this is not the same because of an arbitrary definition of value would not be considered valid. And this would be true for any other video game, and for any other game of chance. Again: any player is free to personally value things differently. But in a discussion of value, the standard definitions must hold, or everyone is just making stuff up randomly.

## Comments

3,091★★★★★Let's say we would get 20,000 shards.

You could get 5 crystals with a 4.5% chance of a 6* (or 95.5% of something else). The chance of not getting a 6* is 95.5%^5 = 0.794%, or 20.6% chance of a 6*.

You could get 10 crystals with a 3% chance of a 6* (or 97% of something else). The chance of not getting a 6* is 97%^10 = 0.737%, or 26.3% chance of a 6*. A slight improvement.

That is, we will be getting the same amount of shards.

4,082★★★★★3,091★★★★★You could get 7 crystals with a 4.5% chance of a 6* (or 95.5% of something else). The chance of not getting a 6* is 95.5%^7 = 72.4%, or 27.6% chance of a 6*.

You could get 14 crystals with a 3% chance of a 6* (or 97% of something else). The chance of not getting a 6* is 97%^14 = 65.3%, or 34.7% chance of a 6*. A slight more improvement.

Overall, the more crystals, the better for us.

323★★★Do we get more Event crystals this month?

1,001★★★★5,625★★★★★2,354★★★★★4,082★★★★★So you choose to forgo the crystals and want to complain about reduced drop rates for what reason?

And you get 15 from the calender so there is no situation that you're worse off here

943★★★★4,082★★★★★33,164★★★★★4,082★★★★★33,164★★★★★33,164★★★★★31★16,503GuardianOnce upon a time, the math to calculate the chance for something to happen or not happen was not common knowledge here on the forums. I'm not sure I would call it common knowledge now, but it is much more commonly discussed. The problem is that it is now being applied universally to all statistical situations whether it actually applies or not.

When we're talking about a situation where a player is attempting to achieve something, one very specific something, that sort of math can calculate the chance of binary success: yes or no. Do I land the featured champ or not? Do I get a 6* champion or not? That sort of thing.

But when it comes to

expected valuethat math doesn't apply, because players do not generally value things based on binary win/loss criteria. One 6* and two 6* champs are both "winning" in such calculations, but they obviously aren't worth the same amount.If we're judging the value of a 4.5% 6* crystal or a 3% 6* crystal that costs half as much, ironically the simple calculation is the correct one. One 4.5% 6* crystal, if all we care about is 6* champs, is worth 1.5 3% crystals. If the 3% crystals cost half as much, then they have more value (this presumes all other things being equal, which may or may not be true depending on the availability of the shards to form them). To put it another way, the expensive crystal offers 50% more 6* champs for 100% more cost. It is not as good of a value.

If I had X number of shards and I had a choice: buy 4.5% crystals at 4000 shards per crystal or buy 3% crystals at 2000 shards per crystal, then assuming I have some multiple of 4k shards (so I'm buying a whole number of each regardless) I'm buying the 3% crystals. They offer the best return.

323★★★There are only 3 facts we know for sure right now

1) The drop rate have been decreased for this month event (we don't know if this is permanent or if it would change back im the next month event)

2) We are getting more Event crystals this month, this is with a little assumption, as we won't know for sure until the event end. (However we also don't know if Kabam will give similar amount going forward, if next Event it drop to 7 Event crystals, in which it would be worst than before).

3) Kabam have not officially announced these changes or plan going forward. So it kind of leave players confused and fearful.

As for my opinion, with so much previous issues of crystals coming out with wrong drop rates (in the pass few months), is it really worth for Kabam to make another set of different crystals rate?

Feel like it just adding more work and info into a systems that is already overloaded data. Feel like extra work for the already overworked development team.

Either way I think it would be helpful to get an official response from the admin team

@Kabam Miike @Kabam Boo @Kabam Zibiit

7,056★★★★★Unless you login (work and time), you don't get it. So it is a free crystal or not.

Your govt send you a voucher to redeem cash at their office. It is free. Do you need to drive a car, walk there or do something?

943★★★★2,550★★★★★A) what we have, and

same number of shards but betters odds at double the shards needed per crystal.

You’re right about the statistics, but it’s a disappointing “oversight” that they failed to mention the crystals reverting to the old rates even though they announced the 2000 shards per crystal.

756★★★16 with a .88 drop rate…

32 with a .52 drop rate…

*trails off into eternity….

3,035★★★★★78★Reading it quickly it gave the me impression that the correct calculation of how many 3% crystals is worth a 4.5% crystal if we want one or more 6* comes simply from the probability ratios p1/p2. Having read some of your posts it's likely I interpreted you incorrectly and you were referring to the cost/return part at the end of the quote (which I agree with). But to avoid any confusions, let’s expand on it.

If you got pass the typos due to writing fast, the probability breakeven by trial numbers I calculated in the previous page for getting at least 1 6* from the two different drop rates was:

N2 = N1 * log(1-p1)/log(1-p2) [1]

Which happens to be close to 1.5 and is very simple math, but it is not coming from p1/p2. Now let’s use the ratio p1/p2 = 1.5 as a condition to see how the expression [1] behaves:

N2/N1 = log(1-1.5p2)/log(1-p2)

Under p1 = 1.5*p2 boundary, N2/N1 has this shape :

N2/N1 match p1/p2 ratio = 1.5 at very low p2 values. From there, the logarithmic functions take over.

As an example

let’s say we pick p2 = 0.5 and p1 = 0.75 (yes, too generous of a crystal, I know). The ratio p1/p2 is still 1.5, but we won’t match the probability of getting at least a 6* until the crystal ratio is 2, or N2 = 2 N1. Going farther means you would need to open more than twice the number of crystals with p2 to even out the chances of opening at least 1 6* from the ones with p1.

Let’s find an independent source to test it: I picked the first link returned googling chance calculator.

If we try and match the number of trials to get equal probability of having one or more for the special case p2 = 0.5 and p1 = 0.75, we get for N1=2:

And we match probabilities when the number of trials satisfy N2 = 2 N1.

78★To summarize my point, while I understand what you say about focusing on the expected value to estimate the crystal equivalence (E[X1] = E[X2]; n1p1 = n2p2 -> n2/n1 = p1/p2), this estimation is informative only if the law of the large numbers apply. That is, if each one of us get to open hundreds of crystals.

It will also work well to control the global resources inflow when changing the drop rates, since it will measure all crystals opened and not a specific account.

But when you have very low sample sizes (like 10-20), the averages you will get will vary wildly from E[X] = np. Also, given how many times I've seen people upset because they have opened, say, 5 crystals and not gotten 1 of something with a 20% chance, in my opinion using the expected value can lead to confusion.

That's why I personally find more informative to compare the probabilities of getting 'at least' 1 6*. And yes, even though getting 1 or 2 or 3 is not the same, considering how low the chances are and the small amount of crystals we get, I would consider 'at least 1' a success.

That being said, the outputs of both approaches are indistinguishable due to the probabilities being very low, so all this text I posted (correct or not) is probably irrelevant and a massive waste of time to anyone reading. Sorry for that!

323★★★The only thing I think I got is that, whether it is better or worst to have lower drop rate with more crystals.

But the key variable that we don't know is will we continue to get more (double the amount) crystals than previous event when the drop rate was higher?

We get more this month, but what is the plan going forward. This where is would be help to get an official response from Kabam

16,503GuardianYou're heading away from the realm of statistics and into the realm of psychology. When we talk about expected value, we mean objective expected value. The value that we can assign mathematically given only that the factors involved are numerically quantified ones. One 6* is twice as valuable as two. If you leave this objective viewpoint you can make anything worth anything, but if you decide to invent an alternate definition of expected value then you'll leave the realm where any other statements anyone makes about value may not apply to you anymore. For example, if you expect the game design to honor that definition of value, that's never going to happen.

You might ask why this definition is the one everyone agrees to, and not some other: in other words, is the definition arbitrary, and any other is equally good. In fact, this definition of objective expected value is the only one that tends to make sense when judging games of chance for a very wide range of reasons. Game theory tends to become nonsensical if you break the definition of expected value. But that's a very extensive discussion.

You're certainly free to say "well, for me all I want is at least one 6*. I don't care if I get one or ten, I want at least one" and then calculate which crystal opportunity satisfies that constraint. Just recognize this isn't how value is generally defined, and thus all situations you're ever likely to find yourself in won't be using it and will likely be highly inconsistent with it.

To summarize, if Kabam makes a crystal with a 20% chance to drop a 6* that costs 4000 shards, and then later makes a crystal with a 10% chance to drop a 6* that costs 2000 shards, from their point of view those two crystals will have equal value (given the same availability of shards) because that's the generally accepted definition of value. Any counterargument that this is not the same because of an arbitrary definition of value would not be considered valid. And this would be true for any other video game, and for any other game of chance. Again: any player is free to personally value things differently. But in a discussion of value, the standard definitions must hold, or everyone is just making stuff up randomly.