First, your original math (the .004 x .004ish) is calculating the chance of pulling one specific champ, then pulling them again immediately. It is a super rare event, but unless you were predicting the event beforehand, just the chances of pulling the same champ is .004 or .4%, which is not that crazy.
You then use two examples of people pulling amazing champions back to back, and only one of a bad champ (psycho man), which doesn’t seem to fit with your next statement of “something fishy” going on. If your friend left the game because two pulls convinced him that he would never grow his roster, then he must not have understood that many people go 10-20 pulls before they get a champ that they really get excited about. Roster growth is based on a large number of champs and a long period of opening.
Third, it is probably more like middle school or elementary maths, but statistics takes some people all the way to college. I’m not sure why your sample size seems to be so small when you say that the “maths aren’t adding up”. In an RNG based game with millions (right?) of downloads, this could easily be happening to hundreds and thousands of summoners without being sketchy. You then say that you’ve been playing for five years but in all those times, have you never pulled the same champ back-to-back?
There’s a 3.6% chance to pull a 6* from these crystals. I’ve personally never pulled 6*s (I’m taking anyone not the same) back to back from those. I have on three separate occasions pulled the same 6* from back to back 6* basics crystals. That math seems very improbable. Back in the day when 6*s champs were first released, and the pool was significantly smaller, I had a friend who pulled beast with his first three 6* crystals.
Do you know how many times a Random number generator can generate the same number again in a row ? The RNG they use in software are Pseudo RNG algorithms. Nothing can be truly random. They tend to generate the same numbers in a row more often than they should, the outcome is more likely to happen that it should. Strange but that's how it is.
This is the real answer and is why you tend to get the same champ in batches of 3+ sometimes when you pop a set of crystals. You won't find the most advanced pseudo RNG algorithm in a mobile game.
The likelihood of getting 2 of the same champs in 2 pulls is not anywhere close to 1:250. If your argument is, I open 1 crystal, then I open a second. Then, yes correct. OP said if I open 2 crystals, what's the chance I get 2 of the same. Completely different formulae.
There’s a 3.6% chance to pull a 6* from these crystals. I’ve personally never pulled 6*s (I’m taking anyone not the same) back to back from those. I have on three separate occasions pulled the same 6* from back to back 6* basics crystals. That math seems very improbable. Back in the day when 6*s champs were first released, and the pool was significantly smaller, I had a friend who pulled beast with his first three 6* crystals.
This literally just happened to me yesterday with the new Cav event crystals. I pulled 6* NT from my first Metal crystal, and 6* Purgatory from my first Women crystal. I’ve never pulled consecutive 6* champs, and my roster is reasonable large.
These are low probability events, either happening a handful of times or not happening to a given player is expected.
The key for me is to not look at it as two individual sets of choices, but as one big choice where the odds don't get "locked in" but more information is given so that you can make a better overall decision. The whole locked-in thing is fine and all, but it doesn't really explain why the odds don't shift, it just pre-supposes that you understand how the giving of information changes the system. I think most people get tripped up because they don't realize that Montii can't actually open the door you chose, and what that does to the probabilities.
This the where educated people tend to go awry. They ask this question: if the odds of the other box keep going up as you open boxes, why don't the odds of *your* box also go up. What's the difference? Your choice makes all the difference, but I don't think it is often explained why your choice box is different from all the other boxes. It is, in a very fundamental way, as you mention: Monty can't open it. But *why* is this important? I'm going to explain this in terms of the "locked in" idea, which is mathematically congruent to your explanation but I believe directly tackles the "why didn't my box change" question more head on.
When Monty opens one of the three doors, or 999,998 empty boxes, or whatever, it is important to remember (as you also mention) he isn't opening randomly. He specifically does not open the option with the prize. So he isn't free to choose any option to open. Equally important, he can't open your choice, because that's structured into the game.
So when Monty starts opening options, there's two options he's not allowed to touch: your choice, and the prize. If you actually chose the prize, then there's actually only one option he can't touch: your choice. But if you didn't choose the prize, then there's two options he can't touch: your choice, and the prize box. To put it another way, the prize protects a box from being opened, just as your choice protects a box from being opened.
The reason why one box's odds improve while the other box's options don't improve is because the last box standing when Monty opens all other options could have been opened. It is a survivor. Why did it survive? Either it was lucky, OR OR OR OR it was never about luck, because it has the prize. The prize box is "protected."
So when the dust settles, we have two options: your choice, and the other one. Yours survived by default: it was guaranteed to survive. So we don't know anything new about your choice. But the other one survived the culling process. We now know something new about the survivor. It *may* have survived because it has the prize, which guaranteed its survival.
Metaphorically, hidden knowledge "flowed" into the survivor box. It didn't flow into your choice box, because the fact that it survived the process tells us nothing.
And that's why one box's odds change, and the other box, the choice box, doesn't. At the end of the game, just before you are allowed to switch, we know more than we did at the start about the survivor box, but know nothing more than we did at the start about the choice box. The choice box's odds are in fact "locked in." The "survivor box's" odds start low, and get progressively higher as Monty opens options, because the more options Monty opens, the more likely it is that the survivor box survived not because it is lucky, but because it contains the prize.
TL;DR: One option survives the game, and survival tells us something about that option. Your choice doesn't really survive anything, and thus we learn nothing new. So the survivor box's odds increase, while the choice box's odds stay the same.
That's a really nice way of describing the Monty Hall scenario.
Do you know how many times a Random number generator can generate the same number again in a row ? The RNG they use in software are Pseudo RNG algorithms. Nothing can be truly random. They tend to generate the same numbers in a row more often than they should, the outcome is more likely to happen that it should. Strange but that's how it is.
This is the real answer and is why you tend to get the same champ in batches of 3+ sometimes when you pop a set of crystals. You won't find the most advanced pseudo RNG algorithm in a mobile game.
This is false (well, the first part is false, the second part is misleading), and it is a fallacy that people keep repeating, even though it is mathematically provably false.
People keep thinking that because "there's no such thing as 'truly random'" the crystal openings must be broken in some obvious way. They aren't. Most of the old school language libraries used a variation of Mersenne Twister as their built in RNG. Mersenne Twister has flaws, but those flaws are only meaningful in very sophisticated circumstances. In the "roll a random number between one and a thousand" scenario, like what you find in a video game's lootboxes, Mersenne Twister contains no flaw that human eyes can detect. It would not generate more three in a rows than statistical chance would dictate.
In fact, it is (essentially) mathematically provable that Mersenne Twister will not generate more three in a rows than statistical chance would dictate for something like MCOC crystals, because Mersenne Twister is a GFSR with a twisting modification to achieve equidistribution of high order bits. To oversimpify a bit, all possible bit sequences (of high order bits) occur the same number of times. Or to put it another way, all possible sequences happen equally often. Three in a row happens exactly as often as random chance would dictate. That's baked into the math.
Modern pRNGs are better than Mersenne, but even Mersenne is sufficiently strong that for something like lootboxes, no human observation could possibly detect any variation from statistical randomness. Mersenne is known to pass most of the Diehard tests for randomness. It isn't going to pass those, and then fail the "MCOC let's open a couple Cav crystals" test.
OP has abandoned this and refused to concede 4 pages of comments ago.
Let this die.
But now we have people conflating “randomly distributed” with “uniformly distributed.” Clusters in random distributions is a whole ‘nother topic we can dive into.
I was going to delve more deeply into the equidistribution properties of FSRs to explain how most modern pRNGs actually satisfy histographic requirements of pRNGs at the expense of eventual periodicity, but I thought that would only add additional confusion.
The whole concept of what randomness even is, is a very complex and subtle one. But it is also mostly irrelevant to the question of how MCOC crystals work, and I wish people who did not actually understand the subject would just stay away from it. For the purposes of a game, crystals do not need to be "truly random" (whatever that means: I don't know anyone who does). They just need to have the following properties:
1. Every option has an equal chance of appearing.
2. For any reasonable length of sequence of openings, all possible sequences are equally likely to occur across the entire playerbase.
3. Future drops are not predictable from observing any reasonable number of prior drops.
4. No factor under the control of the player can predictably influence the opening.
That's what humans colloquially call "random." But it doesn't have to be mathematically random, or entropically random, or any other kind of random. When we talk about random crystals, that's all we care about. And pRNGs can *easily* satisfy these requirements.
The first requirement is obvious. The second one generalizes the first one: the odds of pulling Nebula and then Hercules in sequence is the same as pulling Groot and then Starlord, or Groot and then Groot. So duplicates and triplicates and all other sequences are equally likely. The third says that drops don't follow a predictable pattern.
The last one has an important word people tend to forget. Some people say that crystals are not "random" because pRNGs are predetermined, so in theory a player can influence their drop by picking the time they open the crystal. This is true. But this influence is not predictable. You know that your crystal will generate a different drop if you open it at 10am vs 11am. But you don't know in what way. That makes this influence statistically meaningless.
If MCOC crystals obey those four criteria - you can't predict them, you can't influence them, and everything is equally likely to drop - then they are random enough. Modern pRNGs are random enough.
Do you know how many times a Random number generator can generate the same number again in a row ? The RNG they use in software are Pseudo RNG algorithms. Nothing can be truly random. They tend to generate the same numbers in a row more often than they should, the outcome is more likely to happen that it should. Strange but that's how it is.
This is the real answer and is why you tend to get the same champ in batches of 3+ sometimes when you pop a set of crystals. You won't find the most advanced pseudo RNG algorithm in a mobile game.
This is false (well, the first part is false, the second part is misleading), and it is a fallacy that people keep repeating, even though it is mathematically provably false.
People keep thinking that because "there's no such thing as 'truly random'" the crystal openings must be broken in some obvious way. They aren't. Most of the old school language libraries used a variation of Mersenne Twister as their built in RNG. Mersenne Twister has flaws, but those flaws are only meaningful in very sophisticated circumstances. In the "roll a random number between one and a thousand" scenario, like what you find in a video game's lootboxes, Mersenne Twister contains no flaw that human eyes can detect. It would not generate more three in a rows than statistical chance would dictate.
In fact, it is (essentially) mathematically provable that Mersenne Twister will not generate more three in a rows than statistical chance would dictate for something like MCOC crystals, because Mersenne Twister is a GFSR with a twisting modification to achieve equidistribution of high order bits. To oversimpify a bit, all possible bit sequences (of high order bits) occur the same number of times. Or to put it another way, all possible sequences happen equally often. Three in a row happens exactly as often as random chance would dictate. That's baked into the math.
Modern pRNGs are better than Mersenne, but even Mersenne is sufficiently strong that for something like lootboxes, no human observation could possibly detect any variation from statistical randomness. Mersenne is known to pass most of the Diehard tests for randomness. It isn't going to pass those, and then fail the "MCOC let's open a couple Cav crystals" test.
That'd be great if it was truly rng based. But kabam is a company trying to make money, so algorithms apply. The "overall" drop rate will (has to for platform support) match posted drop rates. However, there is nothing to say that kabam can't cook the books, so to speak, to incentive spenders.
Biggest problem with that mindset, I used to spend. I don't now. 12.0 closed the wallet, spent after dev diaries, focus on the enjoyment. Yeah, fooled me twice.
If you think algorithms and rng are mutually exclusive, they aren't.
For your mathematically impossible... how many characters/rarities of characters are available in a phc? How have I been able to open 4 to 5 phcs many, many, many times. All of them the same character rarity? Anomaly, right. Except not, since it's statistically impossible, or rather so i.probable as to be impossible.
i was teaching probability to my honors kids today and actually used this thread and this example in my class, always enjoy connecting MCOC to my classes haha
i was teaching probability to my honors kids today and actually used this thread and this example in my class, always enjoy connecting MCOC to my classes haha
Do you know how many times a Random number generator can generate the same number again in a row ? The RNG they use in software are Pseudo RNG algorithms. Nothing can be truly random. They tend to generate the same numbers in a row more often than they should, the outcome is more likely to happen that it should. Strange but that's how it is.
This is the real answer and is why you tend to get the same champ in batches of 3+ sometimes when you pop a set of crystals. You won't find the most advanced pseudo RNG algorithm in a mobile game.
This is false (well, the first part is false, the second part is misleading), and it is a fallacy that people keep repeating, even though it is mathematically provably false.
People keep thinking that because "there's no such thing as 'truly random'" the crystal openings must be broken in some obvious way. They aren't. Most of the old school language libraries used a variation of Mersenne Twister as their built in RNG. Mersenne Twister has flaws, but those flaws are only meaningful in very sophisticated circumstances. In the "roll a random number between one and a thousand" scenario, like what you find in a video game's lootboxes, Mersenne Twister contains no flaw that human eyes can detect. It would not generate more three in a rows than statistical chance would dictate.
In fact, it is (essentially) mathematically provable that Mersenne Twister will not generate more three in a rows than statistical chance would dictate for something like MCOC crystals, because Mersenne Twister is a GFSR with a twisting modification to achieve equidistribution of high order bits. To oversimpify a bit, all possible bit sequences (of high order bits) occur the same number of times. Or to put it another way, all possible sequences happen equally often. Three in a row happens exactly as often as random chance would dictate. That's baked into the math.
Modern pRNGs are better than Mersenne, but even Mersenne is sufficiently strong that for something like lootboxes, no human observation could possibly detect any variation from statistical randomness. Mersenne is known to pass most of the Diehard tests for randomness. It isn't going to pass those, and then fail the "MCOC let's open a couple Cav crystals" test.
I know what you're saying but my experience (and no it's not just about confirmation bias) says the RNG is not flawed in the sense that it is rigged, but it is more than expected guaranteed to seed a number in such a way it repeats itself to a point you do notice in each series of events. It's quite common to get 3+ copies of a given champion when opening batches. And that is because the openings are occurring at the same point in time by the same person, and the pRNG is using whatever variables it is programmed to and that influences it to an extent.
The champion will change of course - to me that's the randomness - but the fact that I will get 3 OG Hulks out of 5 PHCs or 4 wolverines out of 6 tells me that there is a seed being used (whatever the pRNG uses, whether it be time of day, my uid, etc) that at that point it time makes it likely to pull the same single number out of the 200 in the crystal several times.
I agree it's not something I can influence and it changes (hence being "random and fair") but it's also less random and noticeable than it would be if I was picking a single pebble out of 200 pebbles in a bag, put it back each time, shaked the bag up and did that 5 more times.
Now personally I don't care myself - i've never started a thread in here complaining about pulls, because I know this is a game and it has software, but at the same time, I don't like the way people say "it's RNG" as if it is perfect randomness and the trend people see (which is basically the same digit rolling up several times out of 200 possible options in a very small set of rolls) doesn't exist.
i was teaching probability to my honors kids today and actually used this thread and this example in my class, always enjoy connecting MCOC to my classes haha
@Glads it's actually kinda ironic you linked that. That part you want to pay attention to is what they call the sum rule.
For the coin flip example the odds of the same 2 results in a row is is 1/2. This is because there is a 1/4 chance of getting 2 heads, a 1/4 chance of getting 2 tails, and a 1/2 chance of getting one of each. You don't care which 2 you get in a row. It could be either tails or heads so you add those probabilities together.
It is the exact same situation with champions. Assuming there are 250 champions in the pool, If you are just talking about the odds of pulling a specific champion twice in a row (such as ghost) it would be 1/250^2. However if you are talking about the odds of pulling any champion twice in a row you would use the sum rule and get something like this 1/250^2 +1/250^2+1/250^2+1/250^2...+1/250^2 for as many champions are are in the pool (we assumed 250 in this case). You could simplify this math to (1/250^2)*250 which is the same as 1/250.
I’m not going to argue with the theory of probability. I’ll just say that I’ve seen different people pull the same champion right on the same minute very often. It just feels like something is off. I guess it’s just a distorted perspective since there are not that many champions. I’m more annoyed about not having pulled a single 6* from cavaliers in the last three events when some in my alliance have without even going for all the available ones. Conspiracy? Nah, probability is a b****.
popped 8 crystals 3 of those were white mags 3*... the odds of that per crystals drop rate is insane and not sure how its possible. people that understand basic math should understand how it shouldnt be happening this often in game
Do you know how many times a Random number generator can generate the same number again in a row ? The RNG they use in software are Pseudo RNG algorithms. Nothing can be truly random. They tend to generate the same numbers in a row more often than they should, the outcome is more likely to happen that it should. Strange but that's how it is.
This is the real answer and is why you tend to get the same champ in batches of 3+ sometimes when you pop a set of crystals. You won't find the most advanced pseudo RNG algorithm in a mobile game.
This is false (well, the first part is false, the second part is misleading), and it is a fallacy that people keep repeating, even though it is mathematically provably false.
People keep thinking that because "there's no such thing as 'truly random'" the crystal openings must be broken in some obvious way. They aren't. Most of the old school language libraries used a variation of Mersenne Twister as their built in RNG. Mersenne Twister has flaws, but those flaws are only meaningful in very sophisticated circumstances. In the "roll a random number between one and a thousand" scenario, like what you find in a video game's lootboxes, Mersenne Twister contains no flaw that human eyes can detect. It would not generate more three in a rows than statistical chance would dictate.
In fact, it is (essentially) mathematically provable that Mersenne Twister will not generate more three in a rows than statistical chance would dictate for something like MCOC crystals, because Mersenne Twister is a GFSR with a twisting modification to achieve equidistribution of high order bits. To oversimpify a bit, all possible bit sequences (of high order bits) occur the same number of times. Or to put it another way, all possible sequences happen equally often. Three in a row happens exactly as often as random chance would dictate. That's baked into the math.
Modern pRNGs are better than Mersenne, but even Mersenne is sufficiently strong that for something like lootboxes, no human observation could possibly detect any variation from statistical randomness. Mersenne is known to pass most of the Diehard tests for randomness. It isn't going to pass those, and then fail the "MCOC let's open a couple Cav crystals" test.
I know what you're saying but my experience (and no it's not just about confirmation bias) says the RNG is not flawed in the sense that it is rigged, but it is more than expected guaranteed to seed a number in such a way it repeats itself to a point you do notice in each series of events. It's quite common to get 3+ copies of a given champion when opening batches. And that is because the openings are occurring at the same point in time by the same person, and the pRNG is using whatever variables it is programmed to and that influences it to an extent.
That's essentially not possible. pRNGs are seeded with entropy sources that cannot causes pRNGs to repeat in the way you describe, except in degenerate ways that would be trivial to spot. If we seed with, say, time, and we (stupidly) reseed with time for every crystal opened, then those crystals would still be opened at slightly different times, and "slightly different" doesn't produce slightly different results: one bit different produces a completely different sequence. Conversely, if we reseed with a fixed value constantly (again, for some dumb reason) then the crystals wouldn't just repeat occasionally, they would get into perfect cycles of repetitiveness which would be wildly easy to spot. I've analyzed a lot of pRNG implementations, in and out of games, and I've spotted many broken implementations. None exhibit the kind of issue you're describing, because such a problem could only happen deliberately. No flaw of implementation I'm aware of can just increase the number of duplicates by a small amount. That's simply impossible.
Besides, when you say it is "quite common" to see 3+ copies of a given champion when opening batches, how common is quite common? How often do you think you're seeing that happen, and how often do you think it is *supposed* to happen? If you think you're seeing a problem, you should have some idea of how much more often it is happening that it is statistically supposed to happen. So these two values are significant: how often do you observe this to happen, and how often do you think you're supposed to see it happen, and what's the difference.
popped 8 crystals 3 of those were white mags 3*... the odds of that per crystals drop rate is insane and not sure how its possible. people that understand basic math should understand how it shouldnt be happening this often in game
to me, i see at least 2x 3x dupe pulls in stacks of 10 popped crystals, every other week. in the alliance feed of crystal openings a lot more often. even to the point where 2 separate accounts pull the same champ back to back of the same star lvl. knowing that i wouldnt put it past kabam to probably just being another bug with coding and people pulling duplicates within the same stacks.
popped 8 crystals 3 of those were white mags 3*... the odds of that per crystals drop rate is insane and not sure how its possible. people that understand basic math should understand how it shouldnt be happening this often in game
to me, i see at least 2x 3x dupe pulls in stacks of 10 popped crystals, every other week. in the alliance feed of crystal openings a lot more often. even to the point where 2 separate accounts pull the same champ back to back of the same star lvl. knowing that i wouldnt put it past kabam to probably just being another bug with coding and people pulling duplicates within the same stacks.
Inputting a pool of 250, and a stack of 10 openings, you get a probability of .1667 of getting a dupe. Basically the same as rolling a die and getting a specific number. You seeing it every other week is probably right on point. You seeing it in your alliance feed much more frequently, when 30 people are opening crystals, makes a ton of sense. What you’re describing is exactly the behavior the math predicts.
Comments
Back in the day when 6*s champs were first released, and the pool was significantly smaller, I had a friend who pulled beast with his first three 6* crystals.
These are low probability events, either happening a handful of times or not happening to a given player is expected.
Any maths teachers in the forums, take note 😉
At one point I pulled him 6 times in a row
Why?
People keep thinking that because "there's no such thing as 'truly random'" the crystal openings must be broken in some obvious way. They aren't. Most of the old school language libraries used a variation of Mersenne Twister as their built in RNG. Mersenne Twister has flaws, but those flaws are only meaningful in very sophisticated circumstances. In the "roll a random number between one and a thousand" scenario, like what you find in a video game's lootboxes, Mersenne Twister contains no flaw that human eyes can detect. It would not generate more three in a rows than statistical chance would dictate.
In fact, it is (essentially) mathematically provable that Mersenne Twister will not generate more three in a rows than statistical chance would dictate for something like MCOC crystals, because Mersenne Twister is a GFSR with a twisting modification to achieve equidistribution of high order bits. To oversimpify a bit, all possible bit sequences (of high order bits) occur the same number of times. Or to put it another way, all possible sequences happen equally often. Three in a row happens exactly as often as random chance would dictate. That's baked into the math.
Modern pRNGs are better than Mersenne, but even Mersenne is sufficiently strong that for something like lootboxes, no human observation could possibly detect any variation from statistical randomness. Mersenne is known to pass most of the Diehard tests for randomness. It isn't going to pass those, and then fail the "MCOC let's open a couple Cav crystals" test.
The whole concept of what randomness even is, is a very complex and subtle one. But it is also mostly irrelevant to the question of how MCOC crystals work, and I wish people who did not actually understand the subject would just stay away from it. For the purposes of a game, crystals do not need to be "truly random" (whatever that means: I don't know anyone who does). They just need to have the following properties:
1. Every option has an equal chance of appearing.
2. For any reasonable length of sequence of openings, all possible sequences are equally likely to occur across the entire playerbase.
3. Future drops are not predictable from observing any reasonable number of prior drops.
4. No factor under the control of the player can predictably influence the opening.
That's what humans colloquially call "random." But it doesn't have to be mathematically random, or entropically random, or any other kind of random. When we talk about random crystals, that's all we care about. And pRNGs can *easily* satisfy these requirements.
The first requirement is obvious. The second one generalizes the first one: the odds of pulling Nebula and then Hercules in sequence is the same as pulling Groot and then Starlord, or Groot and then Groot. So duplicates and triplicates and all other sequences are equally likely. The third says that drops don't follow a predictable pattern.
The last one has an important word people tend to forget. Some people say that crystals are not "random" because pRNGs are predetermined, so in theory a player can influence their drop by picking the time they open the crystal. This is true. But this influence is not predictable. You know that your crystal will generate a different drop if you open it at 10am vs 11am. But you don't know in what way. That makes this influence statistically meaningless.
If MCOC crystals obey those four criteria - you can't predict them, you can't influence them, and everything is equally likely to drop - then they are random enough. Modern pRNGs are random enough.
Biggest problem with that mindset, I used to spend. I don't now. 12.0 closed the wallet, spent after dev diaries, focus on the enjoyment. Yeah, fooled me twice.
If you think algorithms and rng are mutually exclusive, they aren't.
The champion will change of course - to me that's the randomness - but the fact that I will get 3 OG Hulks out of 5 PHCs or 4 wolverines out of 6 tells me that there is a seed being used (whatever the pRNG uses, whether it be time of day, my uid, etc) that at that point it time makes it likely to pull the same single number out of the 200 in the crystal several times.
I agree it's not something I can influence and it changes (hence being "random and fair") but it's also less random and noticeable than it would be if I was picking a single pebble out of 200 pebbles in a bag, put it back each time, shaked the bag up and did that 5 more times.
Now personally I don't care myself - i've never started a thread in here complaining about pulls, because I know this is a game and it has software, but at the same time, I don't like the way people say "it's RNG" as if it is perfect randomness and the trend people see (which is basically the same digit rolling up several times out of 200 possible options in a very small set of rolls) doesn't exist.
This would imply that Real RNG is broken, and the primary reason no one uses it. Reality would be bugged.
For the coin flip example the odds of the same 2 results in a row is is 1/2. This is because there is a 1/4 chance of getting 2 heads, a 1/4 chance of getting 2 tails, and a 1/2 chance of getting one of each. You don't care which 2 you get in a row. It could be either tails or heads so you add those probabilities together.
It is the exact same situation with champions. Assuming there are 250 champions in the pool, If you are just talking about the odds of pulling a specific champion twice in a row (such as ghost) it would be 1/250^2. However if you are talking about the odds of pulling any champion twice in a row you would use the sum rule and get something like this 1/250^2 +1/250^2+1/250^2+1/250^2...+1/250^2 for as many champions are are in the pool (we assumed 250 in this case). You could simplify this math to (1/250^2)*250 which is the same as 1/250.
I guess it’s just a distorted perspective since there are not that many champions.
I’m more annoyed about not having pulled a single 6* from cavaliers in the last three events when some in my alliance have without even going for all the available ones.
Conspiracy? Nah, probability is a b****.
Besides, when you say it is "quite common" to see 3+ copies of a given champion when opening batches, how common is quite common? How often do you think you're seeing that happen, and how often do you think it is *supposed* to happen? If you think you're seeing a problem, you should have some idea of how much more often it is happening that it is statistically supposed to happen. So these two values are significant: how often do you observe this to happen, and how often do you think you're supposed to see it happen, and what's the difference.
Except the pool of champs is smaller than dates in the year, so the probability of finding a pair will be even higher in a given group size.
Inputting a pool of 250, and a stack of 10 openings, you get a probability of .1667 of getting a dupe. Basically the same as rolling a die and getting a specific number. You seeing it every other week is probably right on point. You seeing it in your alliance feed much more frequently, when 30 people are opening crystals, makes a ton of sense. What you’re describing is exactly the behavior the math predicts.