Good point but as a math man you should know that the odds of not pulling a feature should get smaller and smaller as you open more crystals.
Not true I am going to use the coin analogy because many people make this mistake. When you flip a coin and get head since there is a 50/50 chance at both outcomes with your logic the next toss should be more likely to be tails. That is not the case since when you go for the second toss there is still the same 50% chance to get heads. If you get heads 3 more times the next toss is not any more likely to be tails than the last four. The coin does not take the past into consideration and the past flips have no impact on the next. Same with the crystals there is no variable change, past outcomes do not effect the variable and every crystal is individual so for every crystal the variable resets to 20%. This percent chance for a featured does not change so even though it may seem you are more likely to pull the featured the longer you haven''t pulled one mathematically that is not the case, it is random thus chances cannot decrease or increase no matter what the past outcomes were. You made a common mistake most people who are unfamiliar with the maths of probability make so before you judge my mathematical logic please make sure yours is correct.
You are misquoting me to prove yourself correct. If I had 10 shots at a 5* feature, then you better believe I have a better chance of pulling a feature than if I only had 1 crystal. The set of 10 has a better chance than any given 1 crystal.
What you are saying is if I open 9 and haven't received a 5* feature, then I have no better of a chance than the last 9 tries which is true. You are referring to the gamblers fallicy which many people do misunderstand. Just because I have opened 9 doesn't mean I have better or worse odds on the last one.
Perhaps you should improve your reading comprehension skills or troll less on the forums.
Here is some reading material in case you truly believe you are still correct.
Err, what you said was "Good point but as a math man you should know that the odds of not pulling a feature should get smaller and smaller as you open more crystals." The straight forward way to parse that sentence (which is how I parsed it) is that it is saying with each crystal you open, the odds change.
What it sounds like you meant to say is "the odds of not pulling a feature should be smaller if you open a larger number of crystals."
Ah I see that I did say one thing and meant another. That is my bad. Sounds like we were all on the same page this whole time.
1/ Someone has to program the crystal opening rates so it approximates x%.
2/ There should be an audit or check to verify whether the total crystal opening % of the featured champ approximates the % in 1/.
Maybe it is not necessary but it seems logical to collect data to test if x% to obtain featured champ is within the range.
This leads me to ask whether there is an adjustment to the % in order that the overall result will hit x% or thereabouts.
So, when too many people win the featured champ, the % has to be reduced and vice versa.
If this is right, the coin analogy may not be so accurate and not relevant.
While step 2 does seem like a reasonable progression, I don't believe mobile games employ this. As the great Ron Popeil would say, " Set it and Forget it!"
You are misquoting me to prove yourself correct. If I had 10 shots at a 5* feature, then you better believe I have a better chance of pulling a feature than if I only had 1 crystal. The set of 10 has a better chance than any given 1 crystal.
What you are saying is if I open 9 and haven't received a 5* feature, then I have no better of a chance than the last 9 tries which is true. You are referring to the gamblers fallicy which many people do misunderstand. Just because I have opened 9 doesn't mean I have better or worse odds on the last one.
Perhaps you should improve your reading comprehension skills or troll less on the forums.
Here is some reading material in case you truly believe you are still correct.
I did not misquote you in anyway, you said odds of not pulling a featured gets smaller and smaller as you open more crystals. By that wording it the most likely thing you are saying is that there are multiple openings and for every crystal you open the chance gets smaller which I then explained was incorrect like DNA already stated. In my original response I in no way insinuated that what I thought is if you opened say 10 crystals at once you aren't more likely to get a featured than if you opened say two like you apparently were trying to explain to me with your math man response. I was not trolling I was merely explaining the mathematic of probability because many people seem to get that wrong when discussing chances of getting a featured. If you are trying to correct me please at least use correct spelling and grammar in you writing, it makes you seem like you just read up on what gambler's fallacy is and don't actually know what you are talking about, My reading comprehension is fine maybe you could work on your writing skills so you clearly state what you intend to state without chance of confusion due to incorrect wording.
You are misquoting me to prove yourself correct. If I had 10 shots at a 5* feature, then you better believe I have a better chance of pulling a feature than if I only had 1 crystal. The set of 10 has a better chance than any given 1 crystal.
What you are saying is if I open 9 and haven't received a 5* feature, then I have no better of a chance than the last 9 tries which is true. You are referring to the gamblers fallicy which many people do misunderstand. Just because I have opened 9 doesn't mean I have better or worse odds on the last one.
Perhaps you should improve your reading comprehension skills or troll less on the forums.
Here is some reading material in case you truly believe you are still correct.
I did not misquote you in anyway, you said odds of not pulling a featured gets smaller and smaller as you open more crystals. By that wording it the most likely thing you are saying is that there are multiple openings and for every crystal you open the chance gets smaller which I then explained was incorrect like DNA already stated. In my original response I in no way insinuated that what I thought is if you opened say 10 crystals at once you aren't more likely to get a featured than if you opened say two like you apparently were trying to explain to me with your math man response. I was not trolling I was merely explaining the mathematic of probability because many people seem to get that wrong when discussing chances of getting a featured. If you are trying to correct me please at least use correct spelling and grammar in you writing, it makes you seem like you just read up on what gambler's fallacy is and don't actually know what you are talking about, My reading comprehension is fine maybe you could work on your writing skills so you clearly state what you intend to state without chance of confusion due to incorrect wording.
Don't know if I'd throw grammar stones in your glass house, buddy
Good point but as a math man you should know that the odds of not pulling a feature should get smaller and smaller as you open more crystals.
Not true I am going to use the coin analogy because many people make this mistake. When you flip a coin and get head since there is a 50/50 chance at both outcomes with your logic the next toss should be more likely to be tails. That is not the case since when you go for the second toss there is still the same 50% chance to get heads. If you get heads 3 more times the next toss is not any more likely to be tails than the last four. The coin does not take the past into consideration and the past flips have no impact on the next. Same with the crystals there is no variable change, past outcomes do not effect the variable and every crystal is individual so for every crystal the variable resets to 20%. This percent chance for a featured does not change so even though it may seem you are more likely to pull the featured the longer you haven''t pulled one mathematically that is not the case, it is random thus chances cannot decrease or increase no matter what the past outcomes were. You made a common mistake most people who are unfamiliar with the maths of probability make so before you judge my mathematical logic please make sure yours is correct.
actually, you are only partly right. it is true that for the very next coin flip, there is 50% chance of heads.
but even though the odds don't change for the VERY next flip, there are formulas to describe the entire history of flipping coins.
If you have flipped the coin 3 times and have the results of HHH. it is true that each had 50% chance of happening.
however, from the beginning, you are already beating the odds. there is only 1 chance in 8 of getting HHH. and from start to finish, only 1 chance in 16 of getting HHHH.
So, the odds of getting ZERO featured champs in 23 pulls assuming 20% fixed chance for success really is 0.8^23. (1 out of 169.4). If you really want to geek out, look up "Binomial distribution"
No one knows the actual Drop Rate. Some have estimates of 20-25%, but I'm not sure how accurate that is. It's higher than a 4* Featured or a 4* in PHC. That much is probably certain.
0-10 here so I don't buy the 20-25% **** . In total my alliance opened 14 iceman crystals , no one got him
Just cause you don't buy it based of a sample size of 0 doesn't make it true....lot's of people in my alliance got him and I went 2/2, should i assume it's 100% because that's my %?
No one knows the actual Drop Rate. Some have estimates of 20-25%, but I'm not sure how accurate that is. It's higher than a 4* Featured or a 4* in PHC. That much is probably certain.
probably certain. I couldn't agree more...maybe.
Sorry to picking on you, but I think this PERFECTLY sums up the obscurity on drop rates, etc.
Kabam has done a WONDERFUL job of not giving any useful information on this subject, and the community is successfully in the dark.
So as this convo seems to still be going on I just wanna restate the same thing I said early on for the op no one knows take all the post here with a grain of salt NO ONE OTHER THEM SOMEONE WHO WORKS AT KABAM knows
You wonder why Kabam does not give out the rate? I could think one reason; without knowing the rate, people are already complaining about the rate, can you image those people will do if the they go for 0/xx and how they will claim the drop rate is not true?
No one knows the actual Drop Rate. Some have estimates of 20-25%, but I'm not sure how accurate that is. It's higher than a 4* Featured or a 4* in PHC. That much is probably certain.
probably certain. I couldn't agree more...maybe.
Sorry to picking on you, but I think this PERFECTLY sums up the obscurity on drop rates, etc.
Kabam has done a WONDERFUL job of not giving any useful information on this subject, and the community is successfully in the dark.
Well, to be precise, I wasn't asserting anything for sure because I don't know. However, we can deduce that it's higher based on results. The fact is they don't discuss Drop Rates, and that's their prerogative. I'm not entirely convinced it would make a difference. In terms of the data observed, people would still go for the Features they want. The chances of pulling a Featured 5* are higher than pulling one when it hits Basic, based on the numerical possibilities. Either way it's always a chance with RNG because each Crystal is a randomly-generated outcome.
1/ Someone has to program the crystal opening rates so it approximates x%.
2/ There should be an audit or check to verify whether the total crystal opening % of the featured champ approximates the % in 1/.
Maybe it is not necessary but it seems logical to collect data to test if x% to obtain featured champ is within the range.
This leads me to ask whether there is an adjustment to the % in order that the overall result will hit x% or thereabouts.
So, when too many people win the featured champ, the % has to be reduced and vice versa.
If this is right, the coin analogy may not be so accurate and not relevant.
The way this works is that someone programs reward tables that work based on (pseudo)random number generators. Only idiots and geniuses make their own random number generator. Everyone else uses the one that comes with the compiler, generally the C library random function. This function reliably generates random numbers in a range that has already been tested to demonstrate that at least for the purposes of a video game the numbers generated are "random enough." Put simply, "random enough" is when you cannot predict the next number from watching the previous numbers, and every number comes up approximately the same amount of times when averaged over long stretches of random rolls.
To illustrate by example, let's say I'm a game designer (actually, this would be done by a designer, a systems engineer, and a programmer, but ignore that for now). I want to make the featured crystal drop the featured champion 20% of the time, a subfeatured champion 5% of the time, and a basic champion 75% of the time (I'm making reasonable guesses about what the values actually are but I don't know what they are in actual fact). The way I would do this is I would make three reward subtables: the featured table which contains one entry: the featured champ. The subfeatured table would contain all the subfeatured champs. And the basic table would contain all of the current basic champs. Different games do this in different ways. I would then make a featured crystal table that looks like this:
Featured
Featured
Featured
Featured
SubFeatured
Basic
...
Basic
where there are fifteen Basic entries in that table. I then generate a random number from 1 to 20 using the random function. Based on the roll, I look up which subtable to go to and then I roll another random number that chooses between all of the entries in that table. Or I would do this:
Featured 4
Subfeatured 1
Basic 15
using "weights" to weight the rewards. The nice thing about weights is they don't have to add up to 20 or 100 or other "nice" numbers. You could do this:
Featured 7
Subfeatured 2
Basic 25
The weights add up to 34, so the game would roll a random number from 1 to 34 and pick one. You have 7 chances out of 34 to get featured (20.59%), 2 out of 34 to get subfeatured (5.88%) and 25 out of 34 to get basic (73.29%).
In any case, so long as your code is correct and your random number generator is sufficiently random (i.e. you didn't stupidly try to write your own) those rewards will, over long periods of time, come out at the percentages calculated. Because that's how the random generator works. But some games do monitor the actual drop rates using counters so they can tell what the real drop rate is, which will, because of random chance, always be a little different than the calculated values. If it is *really* different, that points to a bug somewhere.
Game devs do not just adjust the random generator like you imply to "force" the numbers to come out to the calculated averages. If it doesn't match, that points to a bug, not a need to alter the random generator. They fix the bug and move on.
The random number generators that are used by modern compilers have been tested to very high degrees to determine their randomness. Things like the Mtwister have been tested to trillions of iterations and more. Kabam might have made an error in their reward tables, but if there's an error in the random number generators used by video games to generate crystal openings, we'd have to play the game for billions of years to have a chance to find that problem.
0-10 here so I don't buy the 20-25% **** . In total my alliance opened 14 iceman crystals , no one got him
Are the people who got two out of two entitled to tell everyone that the odds of opening Iceman are 100%?
Also, I went zero for two, but decided to push for one more opening just before the crystal expired. Got him on try number three. I'm now one for three on featured crystals.
actually, you are only partly right. it is true that for the very next coin flip, there is 50% chance of heads.
but even though the odds don't change for the VERY next flip, there are formulas to describe the entire history of flipping coins.
If you have flipped the coin 3 times and have the results of HHH. it is true that each had 50% chance of happening.
however, from the beginning, you are already beating the odds. there is only 1 chance in 8 of getting HHH. and from start to finish, only 1 chance in 16 of getting HHHH.
So, the odds of getting ZERO featured champs in 23 pulls assuming 20% fixed chance for success really is 0.8^23. (1 out of 169.4). If you really want to geek out, look up "Binomial distribution"
You are forgetting the expectation vs variance in probability theory, binomial distribution gives you the expectation but variance in probability disregards expectation so chances do not increase. Binomial distribution equations require set data before it can calculate probability so there is not room for probability variance, variance which will show when multiple physical trails take place. It all comes down to the regression to the mean, since binomial distribution is set data the regression to the mean is a set gradual pattern while in true trails the regression can correct itself at any given point due to probability variance so it is chances can't increase or decrease due to the fact regression can happen at any time. According to binomial distribution if you flip 20 coins it should be 10 heads and ten tails but in many experiments looking at many different areas probability theory with coin flips this 50/50 result (10 heads/10 tails) happened significantly less frequently than non 50/50 results clearly showing flaws in binomial distribution accurately representing physical results due to its set information. Since players opening crystals is truly random each crystal the gambler's fallacy representation of non increasing probability is correct and past outcomes do not effect future crystal openings. I am a proud geek and am familiar with many of these mathematical concepts.
Here is a fresh idea, a new feature 5* crystal for 50k shards that guarantees the featured champ. I would save for that and I almost never let my shards accumulate to more than 10k. It just takes too long to get that far that I cannot resist the urge to pop the crystal when I do. On second thought, I probably would never make to 50k shards since I don't even have enough discipline to go to 15k, lol.
Here is a fresh idea, a new feature 5* crystal for 50k shards that guarantees the featured champ. I would save for that and I almost never let my shards accumulate to more than 10k. It just takes too long to get that far that I cannot resist the urge to pop the crystal when I do. On second thought, I probably would never make to 50k shards since I don't even have enough discipline to go to 15k, lol.
If our estimates are correct and the odds of pulling featured are 20%, that means the average number of featured crystals you have to open to get the featured is 5. That would mean the cost of a guaranteed featured would have to be more than 5 * 15k = 75k.
I say "more than" because the guarantee is itself worth a premium cost over and above the intrinsic value of the crystal. If a guaranteed crystal existed, I would place its estimated par value at between 100k and 150k fragments.
A "guaranteed" feature 5* Crystal would not exist for shards, the possibility of it would only ever exist in dollars. No saving for guarantees, not allowed 'round here.
actually, you are only partly right. it is true that for the very next coin flip, there is 50% chance of heads.
but even though the odds don't change for the VERY next flip, there are formulas to describe the entire history of flipping coins.
If you have flipped the coin 3 times and have the results of HHH. it is true that each had 50% chance of happening.
however, from the beginning, you are already beating the odds. there is only 1 chance in 8 of getting HHH. and from start to finish, only 1 chance in 16 of getting HHHH.
So, the odds of getting ZERO featured champs in 23 pulls assuming 20% fixed chance for success really is 0.8^23. (1 out of 169.4). If you really want to geek out, look up "Binomial distribution"
You are forgetting the expectation vs variance in probability theory, binomial distribution gives you the expectation but variance in probability disregards expectation so chances do not increase. Binomial distribution equations require set data before it can calculate probability so there is not room for probability variance, variance which will show when multiple physical trails take place. It all comes down to the regression to the mean, since binomial distribution is set data the regression to the mean is a set gradual pattern while in true trails the regression can correct itself at any given point due to probability variance so it is chances can't increase or decrease due to the fact regression can happen at any time. According to binomial distribution if you flip 20 coins it should be 10 heads and ten tails but in many experiments looking at many different areas probability theory with coin flips this 50/50 result (10 heads/10 tails) happened significantly less frequently than non 50/50 results clearly showing flaws in binomial distribution accurately representing physical results due to its set information. Since players opening crystals is truly random each crystal the gambler's fallacy representation of non increasing probability is correct and past outcomes do not effect future crystal openings. I am a proud geek and am familiar with many of these mathematical concepts.
I'm having difficulty parsing this correctly, and it isn't for lack of understanding the mathematical concepts. However:
1. Variance in statistics is a measure of deviation from the mean on a random distribution (technically, the square of the deviation, but that's more technical than I want to get here). I don't think it means what you think it means, because it doesn't have a useful application here (because the random rolls for crystal openings is almost certainly linear, and variance over a linear distribution is mathematically uninteresting).
2. Expectation in probability refers to the mean expected value. Binomial distribution gives you the set of all possible results of a random trial. By definition, binomial distribution (and non-binomial distributions in general) does not give the expectation.
3. Binomial distributions do not calculate the probability of events, they are generated from the probability functions of the trial results.
4. Independent random sequences do not regress to the mean. Regression to the mean only happens when the random variable is itself tied to a mean value. For example, coin flip results do not regress to the mean value, they actually obey what are termed "random walks." Random walks do not preferentially seek a mean. Regression to the mean happens in cases like, say, I play golf and you analyze the scores. My scores aren't mathematically random, there exists an average score that is roughly congruent to my skill level. If I happen to shoot eight under my average one day, there are two possibilities: I got way better, or I happen to get lucky. In this case, you'd expect my scores to "regress to the mean" because it is more likely I got lucky than I got massively better. The randomness to my scores is superimposed upon ("anchored to") a mean value. Scores much higher or much lower are likely to be anomalies and thus the probability is good that the next score will be lower if it was very high or higher if it was very low. That is when "regression to the mean" is a real effect.
5. I think you mean "true trials" not "true trails." And I don't think you're using that term correctly. "True trials" simply refers to probability calculations in which trial results are either true or false (labeled somewhat arbitrarily) where true and false are mutually exclusive and covering (which guarantees that Ptrue = 1-Pfalse).
6. Binomial distribution does not say 20 coin flips will be 10 heads and 10 tails. Binomial distribution gives the set of results as a distribution. My pal WolframAlpha would be helpful here: https://www.wolframalpha.com/input/?i=distribution+of+20+coin+flips cf: Distribution of number of heads.
7. "The gambler's fallacy representation of non increasing probability" is kind of word salad. Delete everything before "past outcomes do not affect future crystal openings" and everything after "past outcomes do not affect future crystal openings" and you have a winner.
Comments
Ah I see that I did say one thing and meant another. That is my bad. Sounds like we were all on the same page this whole time.
While step 2 does seem like a reasonable progression, I don't believe mobile games employ this. As the great Ron Popeil would say, " Set it and Forget it!"
I did not misquote you in anyway, you said odds of not pulling a featured gets smaller and smaller as you open more crystals. By that wording it the most likely thing you are saying is that there are multiple openings and for every crystal you open the chance gets smaller which I then explained was incorrect like DNA already stated. In my original response I in no way insinuated that what I thought is if you opened say 10 crystals at once you aren't more likely to get a featured than if you opened say two like you apparently were trying to explain to me with your math man response. I was not trolling I was merely explaining the mathematic of probability because many people seem to get that wrong when discussing chances of getting a featured. If you are trying to correct me please at least use correct spelling and grammar in you writing, it makes you seem like you just read up on what gambler's fallacy is and don't actually know what you are talking about, My reading comprehension is fine maybe you could work on your writing skills so you clearly state what you intend to state without chance of confusion due to incorrect wording.
Don't know if I'd throw grammar stones in your glass house, buddy
actually, you are only partly right. it is true that for the very next coin flip, there is 50% chance of heads.
but even though the odds don't change for the VERY next flip, there are formulas to describe the entire history of flipping coins.
If you have flipped the coin 3 times and have the results of HHH. it is true that each had 50% chance of happening.
however, from the beginning, you are already beating the odds. there is only 1 chance in 8 of getting HHH. and from start to finish, only 1 chance in 16 of getting HHHH.
So, the odds of getting ZERO featured champs in 23 pulls assuming 20% fixed chance for success really is 0.8^23. (1 out of 169.4). If you really want to geek out, look up "Binomial distribution"
They have their share of bad rolls as well. We only notice the good ones because they broadcast them. Their Drop Rate is no different than ours.
Just cause you don't buy it based of a sample size of 0 doesn't make it true....lot's of people in my alliance got him and I went 2/2, should i assume it's 100% because that's my %?
probably certain. I couldn't agree more...maybe.
Sorry to picking on you, but I think this PERFECTLY sums up the obscurity on drop rates, etc.
Kabam has done a WONDERFUL job of not giving any useful information on this subject, and the community is successfully in the dark.
Y'all can carry on now sorry to interrupt
Well, to be precise, I wasn't asserting anything for sure because I don't know. However, we can deduce that it's higher based on results. The fact is they don't discuss Drop Rates, and that's their prerogative. I'm not entirely convinced it would make a difference. In terms of the data observed, people would still go for the Features they want. The chances of pulling a Featured 5* are higher than pulling one when it hits Basic, based on the numerical possibilities. Either way it's always a chance with RNG because each Crystal is a randomly-generated outcome.
About 20% I'm 4/20 so you have a 1 in 5 chance, which doesn't mean you get one in 5 chances either. I went 1 for 9 on Iceman that really hurt.
The way this works is that someone programs reward tables that work based on (pseudo)random number generators. Only idiots and geniuses make their own random number generator. Everyone else uses the one that comes with the compiler, generally the C library random function. This function reliably generates random numbers in a range that has already been tested to demonstrate that at least for the purposes of a video game the numbers generated are "random enough." Put simply, "random enough" is when you cannot predict the next number from watching the previous numbers, and every number comes up approximately the same amount of times when averaged over long stretches of random rolls.
To illustrate by example, let's say I'm a game designer (actually, this would be done by a designer, a systems engineer, and a programmer, but ignore that for now). I want to make the featured crystal drop the featured champion 20% of the time, a subfeatured champion 5% of the time, and a basic champion 75% of the time (I'm making reasonable guesses about what the values actually are but I don't know what they are in actual fact). The way I would do this is I would make three reward subtables: the featured table which contains one entry: the featured champ. The subfeatured table would contain all the subfeatured champs. And the basic table would contain all of the current basic champs. Different games do this in different ways. I would then make a featured crystal table that looks like this:
Featured
Featured
Featured
Featured
SubFeatured
Basic
...
Basic
where there are fifteen Basic entries in that table. I then generate a random number from 1 to 20 using the random function. Based on the roll, I look up which subtable to go to and then I roll another random number that chooses between all of the entries in that table. Or I would do this:
Featured 4
Subfeatured 1
Basic 15
using "weights" to weight the rewards. The nice thing about weights is they don't have to add up to 20 or 100 or other "nice" numbers. You could do this:
Featured 7
Subfeatured 2
Basic 25
The weights add up to 34, so the game would roll a random number from 1 to 34 and pick one. You have 7 chances out of 34 to get featured (20.59%), 2 out of 34 to get subfeatured (5.88%) and 25 out of 34 to get basic (73.29%).
In any case, so long as your code is correct and your random number generator is sufficiently random (i.e. you didn't stupidly try to write your own) those rewards will, over long periods of time, come out at the percentages calculated. Because that's how the random generator works. But some games do monitor the actual drop rates using counters so they can tell what the real drop rate is, which will, because of random chance, always be a little different than the calculated values. If it is *really* different, that points to a bug somewhere.
Game devs do not just adjust the random generator like you imply to "force" the numbers to come out to the calculated averages. If it doesn't match, that points to a bug, not a need to alter the random generator. They fix the bug and move on.
The random number generators that are used by modern compilers have been tested to very high degrees to determine their randomness. Things like the Mtwister have been tested to trillions of iterations and more. Kabam might have made an error in their reward tables, but if there's an error in the random number generators used by video games to generate crystal openings, we'd have to play the game for billions of years to have a chance to find that problem.
Are the people who got two out of two entitled to tell everyone that the odds of opening Iceman are 100%?
Also, I went zero for two, but decided to push for one more opening just before the crystal expired. Got him on try number three. I'm now one for three on featured crystals.
You are forgetting the expectation vs variance in probability theory, binomial distribution gives you the expectation but variance in probability disregards expectation so chances do not increase. Binomial distribution equations require set data before it can calculate probability so there is not room for probability variance, variance which will show when multiple physical trails take place. It all comes down to the regression to the mean, since binomial distribution is set data the regression to the mean is a set gradual pattern while in true trails the regression can correct itself at any given point due to probability variance so it is chances can't increase or decrease due to the fact regression can happen at any time. According to binomial distribution if you flip 20 coins it should be 10 heads and ten tails but in many experiments looking at many different areas probability theory with coin flips this 50/50 result (10 heads/10 tails) happened significantly less frequently than non 50/50 results clearly showing flaws in binomial distribution accurately representing physical results due to its set information. Since players opening crystals is truly random each crystal the gambler's fallacy representation of non increasing probability is correct and past outcomes do not effect future crystal openings. I am a proud geek and am familiar with many of these mathematical concepts.
If you miss , don't worry . Just keep trying.
If our estimates are correct and the odds of pulling featured are 20%, that means the average number of featured crystals you have to open to get the featured is 5. That would mean the cost of a guaranteed featured would have to be more than 5 * 15k = 75k.
I say "more than" because the guarantee is itself worth a premium cost over and above the intrinsic value of the crystal. If a guaranteed crystal existed, I would place its estimated par value at between 100k and 150k fragments.
I'm having difficulty parsing this correctly, and it isn't for lack of understanding the mathematical concepts. However:
1. Variance in statistics is a measure of deviation from the mean on a random distribution (technically, the square of the deviation, but that's more technical than I want to get here). I don't think it means what you think it means, because it doesn't have a useful application here (because the random rolls for crystal openings is almost certainly linear, and variance over a linear distribution is mathematically uninteresting).
2. Expectation in probability refers to the mean expected value. Binomial distribution gives you the set of all possible results of a random trial. By definition, binomial distribution (and non-binomial distributions in general) does not give the expectation.
3. Binomial distributions do not calculate the probability of events, they are generated from the probability functions of the trial results.
4. Independent random sequences do not regress to the mean. Regression to the mean only happens when the random variable is itself tied to a mean value. For example, coin flip results do not regress to the mean value, they actually obey what are termed "random walks." Random walks do not preferentially seek a mean. Regression to the mean happens in cases like, say, I play golf and you analyze the scores. My scores aren't mathematically random, there exists an average score that is roughly congruent to my skill level. If I happen to shoot eight under my average one day, there are two possibilities: I got way better, or I happen to get lucky. In this case, you'd expect my scores to "regress to the mean" because it is more likely I got lucky than I got massively better. The randomness to my scores is superimposed upon ("anchored to") a mean value. Scores much higher or much lower are likely to be anomalies and thus the probability is good that the next score will be lower if it was very high or higher if it was very low. That is when "regression to the mean" is a real effect.
5. I think you mean "true trials" not "true trails." And I don't think you're using that term correctly. "True trials" simply refers to probability calculations in which trial results are either true or false (labeled somewhat arbitrarily) where true and false are mutually exclusive and covering (which guarantees that Ptrue = 1-Pfalse).
6. Binomial distribution does not say 20 coin flips will be 10 heads and 10 tails. Binomial distribution gives the set of results as a distribution. My pal WolframAlpha would be helpful here: https://www.wolframalpha.com/input/?i=distribution+of+20+coin+flips cf: Distribution of number of heads.
7. "The gambler's fallacy representation of non increasing probability" is kind of word salad. Delete everything before "past outcomes do not affect future crystal openings" and everything after "past outcomes do not affect future crystal openings" and you have a winner.