Understand probability better; drop rate statistics

  • #21
    Uhm the thread is called probability, it's not a given, no discussion of odds, there's no certainty and even if you had a certain Leg drop odds are it isn't the Leg you are looking for and if it is, odds that the stats are perfect are negligible...

    Just enjoy the the game and the adventure...
  • #22
    Quote from Ryude »

    As you can see, even though they are independent events we can increase our chances by simply playing more.

    That part is correct. However, the chances per kill does not increase. There is no such thing as "due". Suppose you killed 500 white and not found a legendary, then you killed another 500 white, the probability of a legendary dropping on these 500 whites does not change. While the chance of no legendary dropping after 1000 kills is lower, this does not take into account that you have already killed the 500. Using conditional probability:

    Chance of no legendaries with 500 white kills: (0.9998)500 = 0.9048283

    Chance of no legendaries with 1000 white kills: (0.9998)1000 = 0.8187144

    Chance of no legendaries with 1000 white kills, supposing that you already know that the first 500 white kills gave you no legendaries: 0.8187144 / 0.9048283 = 0.9048283.

  • #23
    Quote from Ryude

    As you can see, even though they are independent events we can increase our chances by simply playing more.




    That's wrong. Chance (P(X)) and expected value (E(X)) are not the same!
    Last edited by Maragia: 4/16/2014 5:29:52 PM
  • #24
    Quote from Maragia
    Quote from Ryude

    As you can see, even though they are independent events we can increase our chances by simply playing more.



    That's wrong. Chance (P(X)) and expected value (E(X)) are not the same!

    That's why it's called a drop rate. Over a very large number of tries, it will ultimately be correct. This means that if you play enough, you will find a legendary (more than one we hope).
  • #25
    Really its all a matter of perspective. Reminds me of the monty hall paradox. Your perspective before you start playing is different than part way through.

    Using the assumed drop rates mentioned before we could say a few things...
    Lets say you go and decide ok, i'm going to kill 500 monsters, that means i have a ~10% chance of finding a legendary! --this is true.

    Now lets say you go and decide ok, what if i go kill 1000 monsters, that means i have a ~19% chance to find a legendary! -this is also true.

    But if you change your perspective, say you've killed 500 monsters and haven't found anything, you cannot say you have a 19% chance if you go kill another 500. Thats wrong. The next 500 will have a ~10% chance. Once you see the results of those first 500 kills, they are no longer valid as part of your probability for a drop to occur. At the end of the day a given monster, has a given % chance to drop something.

    Regardless how many you've killed before, the next one has the exact same chance to drop a legendary. The probability to drop a legendary being higher from a larger sample is only valid within that given larger sample, if 1000 monsters were killed, what are the chances that a legendary dropped? 19%, but knowing this doesn't mean that the last kill had a greater chance than the first, or next kill. They are all the same.
  • #26
    Quote from Ryude

    That's why it's called a drop rate. Over a very large number of tries, it will ultimately be correct. This means that if you play enough, you will find a legendary (more than one we hope).

    There's no "correctness."

    What you're expressing, in very jilted language, is the Law of Large Numbers. But even that doesn't mean that if the drop rate is 1 every 2 hours that after 100 hours you WILL have 50 in your bag. It means that the more data points you have the closer to that 1:2 ratio you'll observe.

    Furthermore, it does not guarantee that after X hours you'll get a drop, no matter how you attempt to extrapolate it. It doesn't guarantee that if you flip a coin and get tails 50 times in a row that the 51st will be heads. It simply suggests that bad/good streaks, over the long term, are evened out. And there's a huge difference between bad luck being counterbalanced by good luck and "if you play enough, you will find a legendary." They're simply not the same concept. They're not even close to it.
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  • #27
    Quote from shaggy

    What you're expressing, in very jilted language, is the Law of Large Numbers.


    And if you combine the Law of Large Numbers with the psychological concept of Negativity Bias, you pretty much have an explanation for everything that goes on in these threads about drop rates.

    Edit: why is quote markup so goddamn broken on this forum? this post looks perfect when I edit it.
    Last edited by st0rmie: 4/16/2014 10:10:31 PM
  • #28
    As someone said this thread talks about probability not really pure droprates. Keep the discussion going.

    This is what I try to do to temper my own rage:

    • Celebrate/be happy whenever a nice legendary drops!
    • Celebrate/be happy when I optimize my build even more and now do T3 rifts in 15 min! (up from T2 in 20mins)
    I still play 2h every night but now I've increased my chances more, I'm playing 'better.' See how my mood isn't only connected to drops but rather stuff I can control, I'm not purely at the mercy of the 'RNG gods'.
  • #29
    Quote from shaggy
    Quote from BigEd781
    Quote from Kilo55555
    • Time for a Diablo3 example: if you go one hour without a single leg drop, you should not expect to find one the next hour just because of this. What you can say is “after 100 hours I should expect X legs” based on previous calculations.

    Actually, you should. Blizz has a timer in there to account for cold streaks.

    That's misleading at best because you're implying that the "falisafe" mechanism grants you a legendary, automatically, after a certain timeframe. I'm pretty sure that the blues described it as scaling up your chance to find a legendary based on how lucky you are, and never as a guaranteed legendary after X hours.

    So the idea that you could make a statement that after 100 hours you should have X legendaries because of the failsafe is..... misleading at best.

    EDIT
    Why are quotes all jacked up? This new forum software leaves a lot to be desired.



    You can only say that after X hours you should have *at least* Y legendaries. We don't have the numbers, so we have to make an assumption as to how high that bonus scales. If it scales to 100% then we can determine a minimum value.

    And yeah, replies are all jacked.
    Last edited by BigEd781: 4/17/2014 1:22:56 AM
  • #30
    Agree with people that mention BIG numbers. Probability theory is all about it. The more number you have - the more exact result is. Until that RNG brings in total chaos: i.e. yesterday i've got 2 legs in 5 seconds (drop after drop), and then another drop in 3 minutes in same rift - so it was literally a "rain" of legendaries. That doesn't mean I have a magic account which showers me with legendaries every time I play the game.
    Last edited by Valiantshiro: 4/17/2014 1:38:53 AM
  • #31
    Quote shaggy:
    He's the kind of guy who sees five straight "low cards" at a blackjack game, then bursts a vein in his forehead when the dealer pulls a small card and not the ten he needed to break because there's no way that a sixth small card could possibly come out. Even though it's very possible and he's making that statement not based on fact, but based on being very poorly informed.


    Actually that kind of guy has a point in regards to the Black Jack when seeing 5 low cards in a row since those cards has been removed from the pile of probable outcomes (correct me if I'm wrong but isn't that what counting cards is all about?). At the roulette table, if the ball lands on red, it doesn't mean that red is taken out, the ball can land on red again. In Black Jack, the 10 drawn cards are taken out and discarded until the decks are shuffled once more. Of course, he's stupid to think that 5 low cards means a guaranteed 6th high card but it is more likely to get a high card.
    Last edited by Niddro: 4/17/2014 1:33:36 AM
  • #32
    Quote from BigEd781

    You can only say that after X hours you should have *at least* Y legendaries. We don't have the numbers, so we have to make an assumption as to how high that bonus scales. If it scales to 100% then we can determine a minimum value.

    It would be almost impossible for anyone to have luck so bad that their "pity legendary drop chance bonus" scaled up to 100%, though. I mean even at 50% if you kill 10 monsters, statistics tell us that you have a .098% chance of not getting a legendary. That's one hundreth of one percent. If you kill 15 monsters you have a .003% chance of not getting a legendary.

    Quite literally, with the volume of monsters we kill, even a 10% drop chance is virtually a guaranteed legendary, without actually being guaranteed. But the problem is that, statistically, even a 99% drop chance isn't actually guaranteed. And from a logical perspective, it's unlikely that the buff ever scales that high because, as I illustrated before, it's completely unnecessary.

    So determining a minimum value is pretty dicey. I guess we could basically just call the 50% mark a "guaranteed legendary within 50 kills" or something and work from there, but even that isn't really a guarantee, although it's horrendously unlikely that you wouldn't get a legendary at that rate. But you could also make the argument for 30% or even 25%.

    What we do know is that Travis said that 1 every 2 hours is their intended base rate. Some people will get luckier. And there is the failsafe to ensure that you don't get TOO unlucky. So, really, unless someone is averaging significantly less than 1 legendary per 2 hours over an extended period of time, there isn't all that much to discuss.

    And, frankly, if you go 3 hours without a legendary, that kind of "bad luck streak" isn't all that much of an outlier if the baseline is 1 per 2 hours. It *is* unlucky, but it isn't so outrageous that someone should have a heart attack from the shock.
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  • #33
    Quote from shaggy

    [quote=undefined]

    Sure, I'm not trying to figure out what the actual worst case rate would be, but I assume that there is one (which is why I don't buy the "I played for 20 hours and didn't get a leg <sadface>" posts. However, I figure it probably does scale to 100% because, well, why not? If I were writing the code for the bonus after X minutes of no legendary, I would probably write something like:

    bonus = min(100, time_since_leg * some_coefficient);

    There are certainly other cases to be accounted for, i.e., idle time, but that should be the gist of it. Once they get a leg it resets, so why not scale to 100? Otherwise there is still some very small, yet non-zero chance of not getting one for longer than intended. I see no reason to set some cap on the bonus unless they want that small chance to exist for whatever reason. Anyway, this is all rather pedantic at this point.
    Last edited by BigEd781: 4/18/2014 9:55:28 PM
  • #34
    Quote from Niddro

    Actually that kind of guy has a point in regards to the Black Jack when seeing 5 low cards in a row since those cards has been removed from the pile of probable outcomes (correct me if I'm wrong but isn't that what counting cards is all about?). At the roulette table, if the ball lands on red, it doesn't mean that red is taken out, the ball can land on red again. In Black Jack, the 10 drawn cards are taken out and discarded until the decks are shuffled once more. Of course, he's stupid to think that 5 low cards means a guaranteed 6th high card but it is more likely to get a high card.

    Even so, it's just full of inaccuracies.

    An 8-deck shoe has 416 cards. Roughly 2 decks are cut off the end. So that's 104 RANDOM cards that are never played in the shoe, leaving 312 cards left. The 104 cards cut off the back could be all high cards, all low cards, or anywhere in between. So right off the bat we're making assumptions as to the composition of the remaining 312 cards and assumptions typically aren't very good when your money is on the line.

    But let's give some credence to it and assume the 104 cards cut off the back are 100% representative of the composition of a regular deck, basically giving us a 6-deck shoe with zero cards cut off the back. Most players divide cards into 10s/aces and 2-9 in that if they get 5 low cards in a row they expect a face/ace next. While this varies a bit based on the hands (if the dealer is drawing on 16 then, technically a 6 is a "high card"), the safest definition of a "high card" is 10+ since that's what guarantees a dealer bust in the most scenarios.

    In a 6 deck shoe there are 6 * 4 = 24 of each rank. That means there are 24 * 5 = 70 10s and a further 24 aces for a total of 94 "high cards." That leaves 312 - 94 = 218 "low cards."

    The other assumption we have to make is that the cards are shuffled in a way that is semi-uniform. While there can be streaks of small or large cards what you wouldn't expect is all 70 10s to be next to each other. What this implies is that while you might get a +10 scenario or a -10 scenario, you're never going to have a +75 scenario, for example, so the general proportion of cards remains, mostly, intact.

    That being said, if at the start of the shoe, six spots plus the dealer, and of the 13 visible cards, all are 2-9, it would be COMPLETELY FUCKING INCORRECT to say "the hole card must be a 10!" Even with the removal of all those non-face cards, there are still 205 "low cards" versus 94 "high cards." You technically have a slightly higher chance (94 in 299 as opposed to 94 in 312) for that 14th card to be a 10, but it's not very significant. In this EXTREME case there's still a much better chance there's a 2-9 face-down than a 10, J, K, Q, or A.

    For someone to assume that seeing four or five "small cards" in a row would mean that the next card must be an ace or face is simply insanity. It just doesn't work like that, and for most of the shoe even very extreme streaks of low cards in a row don't really sway the odds that much.

    TL;DR on the above
    There are enough cards in a shoe, and enough cards cut off at the end, that unless you're at the last hand or two of the shoe, the fluctuation in "shoe composition" from a streak of four or five small cards in a row isn't enough to really alter anything.

    But, you are technically right that the cards *are* consumed, so it IS more analogous to a roulette wheel. And those roulette players exist. The people who think that five black rolls in a row means that red is "due" and then piss their pants if black comes up again. It's simply faulty logic and applies to ALL games of random chance. In 2009 a woman in Atlantic City held the dice at a craps table for 154 throws. I'm sure that there were people standing there saying how 7 or 11 was "due" but it simply doesn't work like that. Even though the odds of what she did was roughly 1 in 1.54 trillion, it still can happen. No different than black hitting 50 times in a row at a roulette table, or the dealer NOT putting a ace/face on the table for 20 cards at a blackjack table.

    There's a huge difference between "unlikely" and "impossible" and that seems to be what confuses most people and fuels the Gambler's fallacy. And I'll quote Wikipedia:

    The most famous example of the gambler’s fallacy occurred in a game of roulette at the Monte Carlo Casino on August 18, 1913,[5] when the ball fell in black 26 times in a row. This was an extremely uncommon occurrence, although no more nor less common than any of the other 67,108,863 sequences of 26 red or black. Gamblers lost millions of francs betting against black, reasoning incorrectly that the streak was causing an "imbalance" in the randomness of the wheel, and that it had to be followed by a long streak of red.[1]
    Red was "due" in their minds to the point that they lost millions because they lost track of the fact that the 26 previous spins had NO INFLUENCE over the next spin. And this faulty logic seems to be rife in the Diablo community too. "I DID THREE RIFTS AND GOT NOTHING GAME IS BROKE WORLD IS ENDING." Yeah, and red was "due" ... just ask Monte Carlo how much they laughed to the bank over the overall ignorance of the players.
    Last edited by shaggy: 4/17/2014 2:18:01 AM
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  • #35
    People making a post claiming a drop is due will henceforth be known as a Monte Carlo-post ^^
    Last edited by Niddro: 4/17/2014 2:50:19 AM
  • #36
    It's also interesting to note that the longer you play the greater the probability becomes that you will see a "long" streak of 0 legendary drops.

    I have to say I don't know this for certain but I think you can calculate this with a binomial distribution calculation http://stattrek.com/online-calculator/binomial.aspx

    Let's define a "long" streak of 0 legendaries as 4 hours without finding a single legendary item. If we assume the chance of getting a streak of 0 legendaries in 4 hours (or ~20 rifts) is 2.5%.
    That would mean that if you played let's say 120 hours (~ 600 rifts) the chances of seeing exactly one "4 hour streak of 0 legendaries" becomes ~36%. Seeing 1 or MORE "4 hour streaks of 0 legendaries" is about 53%.

    Of course I don't know what the exact chance is that someone finds 0 legendaries in 4 hours. But we can say the chance is more than 0%, considering comments made about the saftey net. So seeing multiple 4 hour streaks is not only possible, it is inevitable as time goes on. This is something people complaining about dry spells need to understand.
  • #37
    Quote from Niddro

    People making a post claiming a drop is due will henceforth be known as a Monte Carlo-post ^^

    QFT. Best thing to come out of this thread. :D
  • #38
    Quote from Kilo55555 »
    Quote from Niddro »
    People making a post claiming a drop is due will henceforth be known as a Monte Carlo-post ^^
    QFT. Best thing to come out of this thread. :D

    I prefer to just call them Gambler's Fallacy posts instead. The word "fallacy" really sticks it into them.

  • #39
    thats an interesting story....is gamblers fallacy strictly the idea that a certain thing is due?

    because if i saw a roulette table hit black 5 or 10 times in a row id be freaking the fuck out....even 15 times or 20 times....an then 25 fucking times....wth?

    I'd react the same way if i saw some kid flip a coin and get heads 26 times in a row.
    or if i saw a royal flush on a professional poker game.....sure that chain of cards has no more chance of being drawn as any other chain of cards. but its still remarkable.

    is any of this type of thinking fool hardy or outlandish? surely I cant be alone here....


    The more i type about this the less it seems to relate to the drop rate of legendary items......but its still interesting.
  • #40
    Quote from Shurgosa »
    thats an interesting story....is gamblers fallacy strictly the idea that a certain thing is due?

    because if i saw a roulette table hit black 5 or 10 times in a row id be freaking the fuck out....even 15 times or 20 times....an then 25 fucking times....wth?

    I'd react the same way if i saw some kid flip a coin and get heads 26 times in a row.
    or if i saw a royal flush on a professional poker game.....sure that chain of cards has no more chance of being drawn as any other chain of cards. but its still remarkable.

    is any of this type of thinking fool hardy or outlandish? surely I cant be alone here....

    The more i type about this the less it seems to relate to the drop rate of legendary items......but its still interesting.

    The gambler's fallacy is indeed thinking that things are "due" just because they have not happened for a while. It is however not the only fallacy out there. It is ok to think unusual things are unusual, but thinking that unusual things implies that the world is out to get you is a fallacy.

    As for a routelette wheel hitting black for 5 or 10 times in a row, there's nothing unusual about it. That's just "RNG". It happens. If it hits 15 or 20, I will start betting black, on the basis that the wheel maybe biased.

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