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Gatcha math and the Featured 6* crystal (not a cry for help. probably)

So I finally ended my run on the current featured 6* crystal today, when I pulled the last thing I was hunting for (OG Scarlet Witch) on pull #69. This is by far the most featured crystals I've ever opened for one version, fueled by my ludicrous pursuit of Scarlet Witch and the fact that 6* shards are just a little too available in a little too many game modes for my own sanity.

Sixty nine crystals sounds like a lot, but it actually isn't a statistically significant amount if you are trying to, say, prove that every drop has an equal chance of falling. You'd need tens of thousands of crystals to do that. It is enough to demonstrate that most of the silly conspiracy theories are just that: silly. Here's my overall distribution of drops:



Out of 69 crystals, 22 were featured drops. The statistical average would be about 17 (69/4). So the crystal isn't significantly biased away from the new featured champs. There's not enough data to prove they are *exactly* balanced, but they are statistically unlikely to be biased low. If I should normally get 17 and I actually got 22, that might be lucky. But if someone thinks they are biased in such a way so that featureds come up, say, half as often as they should, then I got 22 when the crystals are rigged to generate something closer to 8. That's very, very unlikely.

69 crystals isn't enough to show that any one particular drop lands statistically at the presumptive percentage, but is it enough data to show *anything* statistically interesting? As it turns out, it is. We can analyze whether the crystals roughly follow gatcha math. What do I mean by that?

Gatcha mechanics generally refers to the situation where a person/player is confronted with an opportunity to get a random reward, selected from a set of rewards, where the presumptive desire is to collect the entire set (or some desirable subset of the set). Now, most people don't actually try to get all 24 champions in the featured crystal (if for no other reason than most people tend to have some of them already), but what if I analyze the data *as if* I was trying to collect them all. Is there enough data to see if the crystal's random drops roughly follows what I would expect? In at least rough terms, it is.

Let's answer one question first. If you were trying to get at least one of all twenty four champions from the crystal, about how long would that take? This isn't math that is often discussed. You might think you could calculate the odds of pulling the first champ, then the odds of pulling the second champ, and so on. But that doesn't work. While trying to pull the fifteenth champ, you might pull the eighteenth champ first, out of order. Then the math vastly overestimates how long it takes. But as it turns out, there is a relatively simple way to calculate this. We look at the average number of pulls it takes before you pull a champ you haven't pulled yet. If we go step by step, the answer drops out naturally.

We start with no champs and no crystals. How many crystals will it take before we pull a champ we haven't pulled yet? Obviously: 1. Whatever is in that crystal, it is guaranteed to be a new pull. Okay, we have one champion. Now how long will it take to pull a new champ? Well, 23 of the 24 champs would be new, and one would be a dup. If it were 12 new champs out of 24, the odds of pulling a new champ would be 50% - 12/24. And the number of crystals we would need to pull, on average, before we got a new champ would be 2, basically the reciprocal of the odds of a new champ. It is pretty obvious when the numbers are easy. But we generalize this for all situations: the odds of pulling a new champ are X/24, where X is 24 minus the number of new champs you have so far. And the average number of crystals you would need to open to get a new champ is the reciprocal of that, or 24/(1-new).

So to get one new champ it would take 24/24 crystals. To get two new champs would take 24/24 + 24/23 crystals. To get three new champs would take 24/24 + 24/23 + 24/22 crystals, and so on. We get this table:



Now these are all my drops:



And this is a table of how many new champs I had, alongside the statistical average expected:



And how close was my distribution to the predicted gatcha new drop rate?



Pretty close. The crystal behaves, over long stretches, about as you would expect if each drop was roughly identical in drop rate. Now, this doesn't *prove* they are all identical in drop rates. It just shows they behave *like* they would. This means certain kinds of "rigging" are unlikely, in particular while it is statistically possible for some champs to be more likely than average, it is not statistically likely that some champs are far less likely than average. More likely champs would show up earlier, which would show up in the data as the blue line starting off higher than the orange line on the left side of the chart. But champs that were deliberately made much rarer would cause the blue line to drop progressively much lower than the orange line as you went to the right. That doesn't appear to be the case.

In other words, if Groot was twice as likely as every other champ, this sort of analysis wouldn't catch that (although the fact that I only pulled Groot twice would strongly contradict that conjecture). But if, say, all of the featureds were half as likely to show up, that *would* tend to show up here.

The last new champ I pulled was Scarlet Witch, which was the 23rd "new" champ I received from the featured crystals. Statistically speaking, #23 should have arrived on pull #67. It actually arrived on pull #69, which in statistical terms is basically a bullseye. The one champ I never pulled from the featured was Rocket Raccoon. Statistically speaking, I should expect him to show up around crystal #91, or 22 crystals from here (in actuality, I should expect it to take on average 24 crystals from here, because at this point I would be trying to get one out of 24 champs in the crystal, but this opens a different can of statistical analysis worms).

Oh hell no, I'm not opening 24 more featureds to see when Rocket Raccoon shows up.
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Comments

  • OakenshieldOakenshield Posts: 1,132 ★★★
    I knew I was doing significantly better on this crystal than all the other featured I’ve tried - all 7 ‘new’ champs (first time ever), 19 out of 24 champs by my 32nd pull. Now thanks to you I have the math to back it up!

    Thanks for putting the time and effort into this. Will be helpful to link it 1-2 times a day every time there is a new tinfoil hat post on a featured crystal.
  • XFREEDOMXXFREEDOMX Posts: 324 ★★★

    All I took from this is, if you wanted Rocket Racoon, OOF tough luck.

    I wish that was my case! I pulled rocket 6 times out of my 25 featured crystals!

  • dukedevill4412dukedevill4412 Posts: 139
    In over 70 featured and still no QS
  • ReignkingTWReignkingTW Posts: 1,301 ★★★★
    "Sixty nine crystals"

    That's bait, man.
  • Shadow_ShooterShadow_Shooter Posts: 111

    All I took from this is, if you wanted Rocket Racoon, OOF tough luck.

    This would only mean if you’re targeting any SPECIFIC crystal, the odds would’ve been the same.
  • JadedJaded Posts: 5,476 ★★★★★
    Ugh this feature has been painful.

    Og thor and man thing were both sig 20 at the beginning and are now sig 160 each. I still haven’t pulled Valkyrie but I’ve gotten everything else I wanted.
  • CrusaderjrCrusaderjr Posts: 1,059 ★★★★
    RNG, kabam, and the game always know what we want and they will do everything to not give you the resources for rank ups or the champ you want lol
  • Rohit_316Rohit_316 Posts: 3,214 ★★★★★
    I opened 28 featured till now and pulled everything i wanted Titannia , Gorr , Galan ,QS , SW , Hulkbuster but no AA ...he was my 2nd most wanted pull after Galan . Now i gotta save for next featured bit yeah even with me this was the most i have ever gone for any featured .
  • GroundedWisdomGroundedWisdom Posts: 34,895 ★★★★★

    RNG, kabam, and the game always know what we want and they will do everything to not give you the resources for rank ups or the champ you want lol

    Must be legit. I've wanted people to stop posting conspiracies for years, but they keep doing it and I can't level up enough to stop it.
  • CrusaderjrCrusaderjr Posts: 1,059 ★★★★

    RNG, kabam, and the game always know what we want and they will do everything to not give you the resources for rank ups or the champ you want lol

    Must be legit. I've wanted people to stop posting conspiracies for years, but they keep doing it and I can't level up enough to stop it.
    your conspiracy matter as much as the next guys. guess just proves they are all real :#
  • CrusaderjrCrusaderjr Posts: 1,059 ★★★★

    RNG, kabam, and the game always know what we want and they will do everything to not give you the resources for rank ups or the champ you want lol

    DNA must reeealy want a rocket then.
    i had to sell my soul and a left kidney to get SW and archie... so if DNA is doing it the safe way with maths and RNG then he will never get rocket :D
  • SirGamesBondSirGamesBond Posts: 3,455 ★★★★★

    RNG, kabam, and the game always know what we want and they will do everything to not give you the resources for rank ups or the champ you want lol

    DNA must reeealy want a rocket then.
    i had to sell my soul and a left kidney to get SW and archie... so if DNA is doing it the safe way with maths and RNG then he will never get rocket :D
    I assume he already have rocket from the last month's goldpool objective thingy...
    He just didn't pull him from featureds..
  • RNG, kabam, and the game always know what we want and they will do everything to not give you the resources for rank ups or the champ you want lol

    DNA must reeealy want a rocket then.
    Kabam handed me a 6* Rocket during the spin for Rocket event. In a totally random Cav. Because when Yolo decides to rig my crystals, he’s always completely drunk off his &$# when he does it.
  • CrusaderjrCrusaderjr Posts: 1,059 ★★★★

    RNG, kabam, and the game always know what we want and they will do everything to not give you the resources for rank ups or the champ you want lol

    DNA must reeealy want a rocket then.
    i had to sell my soul and a left kidney to get SW and archie... so if DNA is doing it the safe way with maths and RNG then he will never get rocket :D
    I assume he already have rocket from the last month's goldpool objective thingy...
    He just didn't pull him from featureds..
    good point lol, maybe to break the rng curse
  • AverageDesiAverageDesi Posts: 4,775 ★★★★★
    DNA3000 said:

    RNG, kabam, and the game always know what we want and they will do everything to not give you the resources for rank ups or the champ you want lol

    DNA must reeealy want a rocket then.
    Kabam handed me a 6* Rocket during the spin for Rocket event. In a totally random Cav. Because when Yolo decides to rig my crystals, he’s always completely drunk off his &$# when he does it.
    Same . Got him duped from one of those free rocket nexus cavs
  • DrZolaDrZola Posts: 7,762 ★★★★★
    DNA3000 said:

    So I finally ended my run on the current featured 6* crystal today, when I pulled the last thing I was hunting for (OG Scarlet Witch) on pull #69. This is by far the most featured crystals I've ever opened for one version, fueled by my ludicrous pursuit of Scarlet Witch and the fact that 6* shards are just a little too available in a little too many game modes for my own sanity.



    Oh hell no, I'm not opening 24 more featureds to see when Rocket Raccoon shows up.

    One of the things that always amazes me is what I call the “clumpiness” of randomized results over a small sample size. It’s very frustrating, and it honestly is at the root of a lot of the irrational anger at this game’s pRNG engine.

    Look at your results: you have “clumps” of QS, Titania, Hulkbuster and X-23, while Rocket, MrN, Valk, Thor and OGSW barely appear (or don’t).

    Most people understand you are unlikely to get an exactly even distribution. But many don’t get that pRNG will often produce largeish clumps that feel like overweighting or outright error. Add to that the occasional production of crystals with the wrong drop rate and voila—you’ve got a prima facie case of crystal conspiracy.

    I stopped sooner than you at #57. Only Star-Lord and Gorr missing from my pulls, but I did manage 7 Valkyries, 6 Rogues, 6 Titanias and 5 Groots. That’s over 40% of my pulls consolidated across just four champions. If you want to add in the 4 Wiccans, then that’s half of my pulls across five champs. Clumpiness indeed.

    The point is that across such a small number of opportunities clumpiness rules the day. It’s to be expected. Over thousands and thousands of pulls, clumpiness gets smoothed. But not over 57, which is a lot for me individually but not a lot statistically.

    Now, if only the T4c trade-in crystals would stop clumping the opposite way I always need them to…

    Dr. Zola
  • Rudolph_RaindeerRudolph_Raindeer Posts: 202 ★★
    edited December 2022
    Too much text to read, but most likely a good in depth break down.

    So the point being: opening 69 crystals and not getting a specific champ is statistically what you can expect?!
  • Anurag1606Anurag1606 Posts: 1,166 ★★★
    So technically I need to open 69 crystals in next featured crystal to get about 23/24 champs 🤔. Cool. I'm saving and currently at 36 crystals. I'm pretty sure I can make atleast 20 crystals more. Hmm 13 short. Will have to see how that goes. 😊
  • Gogeta91199Gogeta91199 Posts: 987 ★★★★

  • Wong_99Wong_99 Posts: 366 ★★
    DNA3000 said:

    So I finally ended my run on the current featured 6* crystal today, when I pulled the last thing I was hunting for (OG Scarlet Witch) on pull #69. This is by far the most featured crystals I've ever opened for one version, fueled by my ludicrous pursuit of Scarlet Witch and the fact that 6* shards are just a little too available in a little too many game modes for my own sanity.

    Sixty nine crystals sounds like a lot, but it actually isn't a statistically significant amount if you are trying to, say, prove that every drop has an equal chance of falling. You'd need tens of thousands of crystals to do that. It is enough to demonstrate that most of the silly conspiracy theories are just that: silly. Here's my overall distribution of drops:



    Out of 69 crystals, 22 were featured drops. The statistical average would be about 17 (69/4). So the crystal isn't significantly biased away from the new featured champs. There's not enough data to prove they are *exactly* balanced, but they are statistically unlikely to be biased low. If I should normally get 17 and I actually got 22, that might be lucky. But if someone thinks they are biased in such a way so that featureds come up, say, half as often as they should, then I got 22 when the crystals are rigged to generate something closer to 8. That's very, very unlikely.

    69 crystals isn't enough to show that any one particular drop lands statistically at the presumptive percentage, but is it enough data to show *anything* statistically interesting? As it turns out, it is. We can analyze whether the crystals roughly follow gatcha math. What do I mean by that?

    Gatcha mechanics generally refers to the situation where a person/player is confronted with an opportunity to get a random reward, selected from a set of rewards, where the presumptive desire is to collect the entire set (or some desirable subset of the set). Now, most people don't actually try to get all 24 champions in the featured crystal (if for no other reason than most people tend to have some of them already), but what if I analyze the data *as if* I was trying to collect them all. Is there enough data to see if the crystal's random drops roughly follows what I would expect? In at least rough terms, it is.

    Let's answer one question first. If you were trying to get at least one of all twenty four champions from the crystal, about how long would that take? This isn't math that is often discussed. You might think you could calculate the odds of pulling the first champ, then the odds of pulling the second champ, and so on. But that doesn't work. While trying to pull the fifteenth champ, you might pull the eighteenth champ first, out of order. Then the math vastly overestimates how long it takes. But as it turns out, there is a relatively simple way to calculate this. We look at the average number of pulls it takes before you pull a champ you haven't pulled yet. If we go step by step, the answer drops out naturally.

    We start with no champs and no crystals. How many crystals will it take before we pull a champ we haven't pulled yet? Obviously: 1. Whatever is in that crystal, it is guaranteed to be a new pull. Okay, we have one champion. Now how long will it take to pull a new champ? Well, 23 of the 24 champs would be new, and one would be a dup. If it were 12 new champs out of 24, the odds of pulling a new champ would be 50% - 12/24. And the number of crystals we would need to pull, on average, before we got a new champ would be 2, basically the reciprocal of the odds of a new champ. It is pretty obvious when the numbers are easy. But we generalize this for all situations: the odds of pulling a new champ are X/24, where X is 24 minus the number of new champs you have so far. And the average number of crystals you would need to open to get a new champ is the reciprocal of that, or 24/(1-new).

    So to get one new champ it would take 24/24 crystals. To get two new champs would take 24/24 + 24/23 crystals. To get three new champs would take 24/24 + 24/23 + 24/22 crystals, and so on. We get this table:



    Now these are all my drops:



    And this is a table of how many new champs I had, alongside the statistical average expected:



    And how close was my distribution to the predicted gatcha new drop rate?



    Pretty close. The crystal behaves, over long stretches, about as you would expect if each drop was roughly identical in drop rate. Now, this doesn't *prove* they are all identical in drop rates. It just shows they behave *like* they would. This means certain kinds of "rigging" are unlikely, in particular while it is statistically possible for some champs to be more likely than average, it is not statistically likely that some champs are far less likely than average. More likely champs would show up earlier, which would show up in the data as the blue line starting off higher than the orange line on the left side of the chart. But champs that were deliberately made much rarer would cause the blue line to drop progressively much lower than the orange line as you went to the right. That doesn't appear to be the case.

    In other words, if Groot was twice as likely as every other champ, this sort of analysis wouldn't catch that (although the fact that I only pulled Groot twice would strongly contradict that conjecture). But if, say, all of the featureds were half as likely to show up, that *would* tend to show up here.

    The last new champ I pulled was Scarlet Witch, which was the 23rd "new" champ I received from the featured crystals. Statistically speaking, #23 should have arrived on pull #67. It actually arrived on pull #69, which in statistical terms is basically a bullseye. The one champ I never pulled from the featured was Rocket Raccoon. Statistically speaking, I should expect him to show up around crystal #91, or 22 crystals from here (in actuality, I should expect it to take on average 24 crystals from here, because at this point I would be trying to get one out of 24 champs in the crystal, but this opens a different can of statistical analysis worms).

    Oh hell no, I'm not opening 24 more featureds to see when Rocket Raccoon shows up.

    It’s sad that you didn’t get Rocket, I’ve opened two crystals and pulled Rocket both times. Hahaha
  • GroundedWisdomGroundedWisdom Posts: 34,895 ★★★★★

    RNG, kabam, and the game always know what we want and they will do everything to not give you the resources for rank ups or the champ you want lol

    DNA must reeealy want a rocket then.
    i had to sell my soul and a left kidney to get SW and archie... so if DNA is doing it the safe way with maths and RNG then he will never get rocket :D
    Funny. I pulled SW and Duped my AA in the last week or so. Pulled Rintrah and Herc as well.
  • DemonzfyreDemonzfyre Posts: 19,574 ★★★★★

    RNG, kabam, and the game always know what we want and they will do everything to not give you the resources for rank ups or the champ you want lol

    Weird. Their algorithm must be broken. I got a skill R3-R4 gem from 8.1. I pulled and duped Valk over the course of 3 featureds.
    I also pulled and duped all 6 featured champs except QS. I pulled him but probably won't open more because o want iDoom.

    You should unsubscribe from Prof Hoff.
  • DrZolaDrZola Posts: 7,762 ★★★★★
    Pikolu said:

    Wong_99 said:

    DNA3000 said:

    So I finally ended my run on the current featured 6* crystal today, when I pulled the last thing I was hunting for (OG Scarlet Witch) on pull #69.
    Oh hell no, I'm not opening 24 more featureds to see when Rocket Raccoon shows up.


    It’s sad that you didn’t get Rocket, I’ve opened two crystals and pulled Rocket both times. Hahaha
    Protip: on long posts, don't quote the whole thing. We all know you are talking to OP without the quote
    The ellipsis. It is your friend.

    Dr. Zola
  • DrZola said:

    DNA3000 said:

    So I finally ended my run on the current featured 6* crystal today, when I pulled the last thing I was hunting for (OG Scarlet Witch) on pull #69. This is by far the most featured crystals I've ever opened for one version, fueled by my ludicrous pursuit of Scarlet Witch and the fact that 6* shards are just a little too available in a little too many game modes for my own sanity.



    Oh hell no, I'm not opening 24 more featureds to see when Rocket Raccoon shows up.

    One of the things that always amazes me is what I call the “clumpiness” of randomized results over a small sample size. It’s very frustrating, and it honestly is at the root of a lot of the irrational anger at this game’s pRNG engine.

    ...

    The point is that across such a small number of opportunities clumpiness rules the day. It’s to be expected. Over thousands and thousands of pulls, clumpiness gets smoothed. But not over 57, which is a lot for me individually but not a lot statistically.
    That "clumpiness" which is baked into the random number generators (and randomness in general) is actually a desirable property of random rewards. It ensures that across the entire playerbase, most players will get different results. If the crystals were more "even" in nature, then that would mean all players who opened the same number of crystals as I did would basically have the same drops as I did. "Even" crystals generate far more homogenous results, and you end up with a far more homogenous player base. Random rewards are in the long run even, but in the short run extremely uneven, which means everyone gets different results, which is something game operators actually want. When everyone is the same, the game gets a lot more boring.

    Player metas are a very powerful homogenizing force. Players who don't know any better follow the leader. They rank up who they are told they should rank up. If everyone got the same drops, 99% of the players would probably have literally identical rosters. Random champion acquisition forces players to try to make due with what they get. It pushes back somewhat on homogenizing metas. If players start off with differing rosters, they might hold onto that deeper into their playing lives knowing they don't have to do what the common meta ideas recommend.

    "Clumpy" randomness is one way to look at it: by looking at what players get a lot more of. But "Gappy" randomness is the other way to look at it: focusing on what players don't get precisely because they got more of other things. It is the combination of the clumps and the gaps that give each player a slightly different experience than most other players. This is very, very deliberately baked into why champion acquisition is random.
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