Submitted by Reason-Local t3_11de5ag in explainlikeimfive
johrnjohrn t1_ja8wxcq wrote
I've understood the math on this for a long time, but can someone here help me understand....... if you rolled a 6 on 1,000 rolls simultaneously, aren't you forced at some point to grapple that the odds of that streak are incredibly, incredibly low, and the likelihood that it will be broken is a fair bet to take on that information alone?
Like, if someone said before the rolls "here, I'll give you a billion dollars if that dice rolls 6 1000 times", you inherently know that that is highly unlikely to happen.
I get it, each roll is independent, but the "gamblers fallacy" doesn't seem like complete fallacy in this scenario. Someone please help me close the gap on my understanding.
I'm guessing someone is going to answer, "You have the same chance regardless of 1,000 rolls". I know. But in a way you're betting now with historic information that 1,001 rolls in a row is likely enough to bet on it.
If 1,000 isn't enough to get us past the "independent roll" answer, let's go with one trillion. One quintillion rolls. I don't care. At some point you would question the validity of probabilites and all that, right?
totalrefan t1_ja9cnke wrote
If a die rolled the same 1000 times in a row, than practically it is far more likely that the die is rigged. But the way that probability theory is discussed wouldn't account for such a thing that isn't specifically mentioned in the problem presented. A real life situation has many more variables than a hypothetical one.
Monimonika18 t1_ja91klv wrote
Because "not 6" is 5 times more likely to happen than 6 on a six sided die in a single throw. You're focusing on the one combination result that is 6 for all 1000 dice rolled and comparing it to the many combinations that are not that one single very specific result you want.
If the billionaire gave you a die to roll 1000 times to get a single specific order of numbers like "123545453621145..." you'd still have the same horrible odds of getting that one result.
Now if it were 1000 die to roll and the result must be all 6s except one single roll anywhere among the 1000 rolls, your odds of getting that goes up because there are more than 1 results that match that criteria.
Make it must get only one 6 anywhere out of 1000 throws/die while the rest are not 6 and your odds are much better because there are even more results that match that criteria.
Going back, if you had already thrown all 6s for 999 throws and you only have to throw the last die, your chances are 1/6 because the result you want is 1 out of 6 results now. The more 6s you got, the probability of getting the rest of the throws as 6s went up because there are less possible results available for the rest of the row of throws. Up until you achieve all 6s, at which point the probability is now 1 out of 1.
johrnjohrn t1_ja92xlc wrote
Indeed, you have repeated the part I already understand, but it feels like saying that, "The probability that I could transform into a duck are the same in this moment that they were 10 seconds ago according to quantum physics". At some point it seems that a certain number of throws in a row would force us to consider things differently. If I did indeed turn into a duck, folks would not shrug that off as a quantum possibility, even if highly improbable. Hypothetically, if you had one quintillion throws in a row, you would have a team of scientists on the scene and it would make international news. Nobody would ever shrug that off and say "ah well, the probability of the next roll doesn't change." At that point, all involved scientists and statisticians and any interested parties would effectively be falling victim to the gamblers fallacy, but it still seems to make sense that they would, right?
atheism-blocker101 t1_ja99500 wrote
I feel bad for you because I 100% think the exact same way as you do about this and am second-hand frustrated that nobody has dealt with your actual point
johrnjohrn t1_ja9a36l wrote
Thanks for validating. I empathize with the math. When it comes to casinos or whatever, there are likely strings that will come out of any number of dice throws, but you bet your ass that at some point the pit bosses will start watching closely, never dismissing the lead up to this point. At some point the manager is called in from off duty. At some point the game is shut down. The gambler can't just get away with saying, "But each roll has an independent probability!" No, the casino crew has now "fallen victim to the gamblers fallacy". But inherently, we understand they haven't. They have made a reasonable decision that they can't afford that person throwing the dice one more time. But what was the point of shutting it down if the dice were fair and any number of gamblers could start doing the same thing on any other number of tables? Do all casinos just shut down forever, therefore falling victim to some version of the fallacy?
RedFiveIron t1_ja9b547 wrote
If the die is fair the odds don't change. The historical information is irrelevant as the die has no memory.
A long string of rolling the same number may indicate the die is not fair, in a non-hypothetical situation.
johrnjohrn t1_ja9cn4k wrote
Would you personally brush off the one quintillion consecutive throws if the die was determined to be fair by a team of scientists? Then another quintillion? And if someone now gave you the chance to bet on the next outcome, which would you choose? I argue that if you are rational you would bet that the streak continues. But mathematically you shouldn't change your bet, and you should ignore the two quintillion consecutive throws up until this point, right? Do you see the problem here?
RedFiveIron t1_ja9dqez wrote
If the die is fair then it's fair, that's all she wrote. Rationally you should always follow the math. Constructing an absurdly unlikely scenario doesn't invalidate the math.
johrnjohrn t1_ja9eo35 wrote
What does it being absurdly unlikely have to do with anything? And at what point do you consider it "absurdly unlikely"? 100 throws? 1000 throws? 1 million throws?
By even pointing out the absurdity seems to indicate that you have an inherent understanding that it does at some point begin to matter.
RedFiveIron t1_ja9g8f3 wrote
You're constructing an extremely unlikely scenario to rationalize thinking a die has memory. It does not.
The previous results don't affect future outcomes for a fair die, no matter what those previous results are or how unlikely that outcome was.
Let me toss that back at you: How many unlikely outcomes have to occur before it "begins to matter"? Is one enough to start ignoring the math? Ten? A thousand?
johrnjohrn t1_ja9qjlw wrote
I never offered that the die has memory. I only offered a hypothetical in which a fair die rolled one quintillion times on the same number by what a mathematician would say is pure chance. And your suggestion that I believe that implies that you have some inherent belief that the math "breaks down" at some undefined, seemingly ridiculous point. Regardless of the number of rolls I pick, you will say it doesn't matter and I say at some point it does. That is the rub. You have absolute confidence that any limitless number I can think of wouldn't sway you into reconsidering which number would be most "rational" to pick next if this all occurred in front of your eyes.
I think what I'm really saying is that normally we'd expect, on average, a die that may roll the same number that can be explained with mathematical probabilities. And those probabilistic averages play out the same, all day every day in casinos everywhere, because we observe them, and the laws of physics appear fixed. Any gambler who thinks those laws of physics and probabilities will change based on their crude observations of a small number of rolls is, in fact, a fool.
Now, you suddenly have an outlier that outlies averages so far that the whole casino industry topples because of it. Although my scenario is absurdly unlikely, your math shows that it is equally possible, albeit unequally probable. Is the gambler who watches the seemingly supernatural phenomenon unfold in real time all that foolish if they were to bet on the next outcome to be the same as the prior quintillion?
I suppose this might be a question of philosophy and not math. And I'm not arguing with the defined math, but I firmly stand beside the point that eventually it is not irrational to assume the same number might be rolled one more time after observing it a quintillion times.
cjo20 t1_ja9nht3 wrote
The “absurdly unlikely” part comes in to play in being able to view the events from two different perspectives.
One is that, if someone claimed at the outset, that they could predict the next quintillion rolls of the die (whatever values those might be), the probability of all of them being correct is vanishingly small - each of the 6^quintillion combinations almost certainly won’t show up, only one will, and you’re relying on picking that one sequence.
However, once you’ve correctly predicted the quintillion rolls in a row, if you then say “I’m going to roll a 6 next”, you aren’t any more or less likely to get it right than you were on the first roll.
The probability of being able to predict (N+1) correct dice rolls is N * 1/6.
1 roll: 1/6
2 rolls: 1/6 * 1/6 = 1/36
3 rolls: 1/36 * 1/6 = 1/216 Etc.
If you’ve already done the N dice rolls, you’ve already dealt with the probability of getting to where you are in the chain of rolls. The probability to advance to the next step in the chain is always the same though, even if the chances of you successfully getting to that point in the chain are infinitesimal. You’d still expect 5/6 to get it wrong at the next roll, 35/36 to get it wrong in the next 2 rolls, and 215/216 to get it wrong in the next 3 rolls.
johrnjohrn t1_ja9oihv wrote
You have done a good job of explaining the math. And thank you.
Now, you're sitting at that gambling table and someone gives you the opportunity to choose one number that will appear on the die for the next roll after one quintillion. Are you going to choose some number other than the one that came up one quintillion times or some other number? I imagine you would choose the same number instead of picking some other number at random.
cjo20 t1_ja9p3u1 wrote
Ultimately, it doesn’t matter which number I pick, I’ve still got the same chance of being right as with any other number.
People can use superstition to help them decide, but it doesn’t make them any more likely to be right. Some people will choose 6 because “that’s got to be right, it’s happened so often”. Others will choose their favourite number “because that’s lucky”. Others will choose anything but 6 because “they can’t be that lucky”. Any logic you try and apply to it to say “this outcome is more likely than any other” is just your brain tricking you.
johrnjohrn t1_ja9s99d wrote
Would you not sit up at a story on the news where someone rolled 7's at a craps table for a year straight, only stopping to eat and sleep? If you paid any attention to that story would you be a fool? They bring officials in and claim the game is still fair and allow it to continue. Are you a fool if you claim it is rigged? Now that same roller rolls for multiple decades. Do you still calmly say, "we are foolish to assume this person will roll 7's one more time just because of the past 50 years they have continued to roll 7's. Each roll is a new roll." ?
cjo20 t1_ja9trp6 wrote
If you’re trying to construct an actual scenario, a casino wouldn’t let that happen. They’d kick the player out because “they believe them to be an advantaged player”, because they don’t like losing money. And eventually you reach a point where it’s simply more likely that there is a bias somewhere in the system that hasn’t been detected yet.
That means it would be a feature of the system (player / table / dice) rather than of the maths - the maths is based on perfectly controlled probabilities.
Practically, you can’t ensure it’s a 100% fair system, so the simple “each outcome is 1/6” breaks down. If you could guarantee that it was perfectly fair, then what I said earlier stands. In a Real-World situation, the assumptions change significantly - you can’t have perfect knowledge of everyone’s intentions, whether it could be a scam etc.
EDIT: however, most gamblers fallacies aren’t based on the idea “I have actual evidence that the system is rigged”. Things like “5 hasn’t come up on the roulette wheel, it must be overdue” aren’t based on an assumption of bias, they’re based on an assumption of fairness, which says that eventually all numbers will come up equally. However, they don’t have to come up equally before the heat death of the universe.
johrnjohrn t1_ja9vmsv wrote
I'm not trying to construct an actual scenario. I am constructing a hypothetical scenario that says there is no chance that the system is rigged, and there are a quintillion throws that are all identical, which is entirely possible, but highly improbable. In real life we can say, "that would never happen", but the math says you are incorrect and it 100% could happen. Now, this situation, which is mathematically possible, plays out (hypothetically). Which bet are you going to make after the one quintillionth throw? And are you a fool if you use past information to say the next throw will remain the same as the past quintillion?
cjo20 t1_ja9x1ev wrote
Again, if it’s guaranteed to be mathematically exactly fair, then by the maths I posted earlier, claiming you have better than 1/6 chance of getting the next one right is mathematically impossible, by definition.
To be clear: you’re defining a situation whereby you are guaranteed to only have a 1/6 chance of getting the next number correct, whichever you pick, and then saying “isn’t it better to stick with the number that came up before?”. Simply, no, it’s not, because of the way you defined the system.
Monimonika18 t1_jadhi4i wrote
Thanks for pointing out that commenter's moving goalposts.
Nerdloaf t1_ja9tszy wrote
The poster is conflating two different and unrelated things. Determining whether a coin is fair, or putting a probability on it being fair after observing a number of tosses has very little to do with “if I just got three heads in a row with a fair coin, what’s the probability of getting a fourth?”.
johrnjohrn t1_ja9fnnh wrote
To add to this, maybe what I'm touching on without knowing it is similar to the problems Einstein was trying to solve when euclidian geometry failed. Yes, space bends as does time when you get off the paper and into the real universe. Probablities that explain away the gamblers fallacy as a fallacy maybe break in the real world when pushed to the same brink as Einstein pushed things when he invented relativity. Maybe??
hero_in_time t1_ja9s7y5 wrote
In a universe of infinite time and space, everything is probable.
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