Each time you roll the six-sided die, there's six possible outcomes: 1, 2, 3, 4, 5, 6. The probability of each one of those is 1/6 (16.66...%), and the total is 1 (100%). It doesn't change with each roll, because you aren't changing the die, or what you are doing. There's nothing connecting one roll to the next; they are completely independent.

The probability of rolling a six on any fair die roll does not change. It will always be 1/6. The idea that the probability goes up if you haven't rolled a six in a while, or down if you just rolled a six, is known as the gambler's fallacy.

Coin toss is an easier example. The coin always has a 50 percent probability that it will land heads or tails on any individual toss.

But when you observe how often a coin will land heads N number of times in a row, we see that as N increases, the probability of success decreases. This is because as you toss coins, each toss brings with it the collective probabilities from all previous tosses.

So while each toss always has a fifty/fifty chance of landing heads, trying to predict how often a particular number of heads in a row will occur depends on how many coins you intend to toss, and the probability that each toss will land on a particular side.

In the scenario you gave, the reason that it doesn’t increase the probability is the because the other sides don’t go away after rolling. For example, if you roll a 10 on a sixteen sided die, that 10 still stays on the die for any subsequent roll attempts, so the amount of sides stays the same, and each number has a 1/16 chance of being rolled.

Probability can change in certain situations, for example a raffle. If 200 names are put in a hat, and one Is drawn, that name is now removed, and your name has now got a 1/199 chance of being drawn. Make sense?

Because the dice doesn't remember what's happened in the past. It's an inanimate object that doesn't undergo any significant change after being rolled.

Can you imagine if the opposite was true? If dice remembered the past and tried to correct for it? How would you know if you're holding a normal dice or dice that (unbeknownst to you) has recently rolled 6 100 times and is now determined to roll something different?

Probability (and science in general) hinges heavily on semantics. The chance to roll a 6 on a die never changes (1/6), but the chance of having 10 consecutive non-6 rolls is relatively low ([5/6]¹⁰=.162), so the chance of rolling exactly one 6 in a string of 10 rolls is 1-.162=0.838. So if you've had nine non-6s it feels like there is an ~84% chance of rolling a 6 on the next roll, but it is still just a ~17% chance.

You are mistaking this for the probability for a set of events. An event in this case is a roll on a dice - 1/6th chance to roll a number. The chance never changes. BUT the chance of rolling any other number decreases the more you add the events into the set. But that's the probability for the whole set.

Set of 1 roll is 5/6 (times 100 if you wanna see it in % ) which is roughly 83,33 %

But set of 10 rolls (any other than 6) is 5/6 ^ 10 which is roughly 16,15 %

Now from this it seems like the chance is getting bigger, but what's changing is the set you're tracking probability for. The rolls are still 1/6th for a number :)

edit> to give you maybe a better example considering my English and explanation skills :D

if you roll 9 times any other number than 6, then the next roll being any other number while considering the whole set is 16,15 % and the roll for the 6 thus increased to awesome 83,85 %! But that's ONLY because we make the probability based on the previous rolls. The chance the next roll will be any other number on its own is still 5/6 and the chance for the 6 is still 1/6.

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?

No, there isn't. Every die roll is an independent event. Thus, the probability of rolling any number is the same no matter how many times you roll it. The likelihood of roll a 6 100 times in a row is very low, but it can happen. The next time you roll, the chances of any number coming up are the same as they were before.

The odds aren't per side, the odds are any side. Probability would only increase if after each roll, one side of the dice was eliminated. Other than that, the odds remain the same.

The answer is simple, how would the dice remember what you rolled before? Each roll is an independent event, there's no memory of the last one that affects the next one.

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The problem with probabilities is always, imo, semantic. We call the unknown "probability", for lack of a better way to describe things. But nothing is random, in reality. That's where the issue lies.

If I roll a die, I don't really have 1/6 chance of rolling a 6. I have either 0% chance, or 100% chance - depends on how I roll it. If I roll it the same way every time, I'll get the same result every time. Rolling a dice isn't "random". We call it random, because we don't know HOW we rolled it.

As for why this thing we call probability doesn't change, that's the easy part. A die has 6 faces, and if we could toss it randomly (we can't, randomness doesn't exist), any of them can come up, and they'll come up the same number of times (as in, given a truly random toss, any of them can come up as much as any other). So we say if we roll a d6, each outcome happens 1/6 of the times. Doesn't matter if you rolled it once, or 1000 times - each roll can come up as any number 1/6 of the time.

Ask yourself this—does the die remember what it previously landed on? If not, how could the odds go up? It’s a physical thing that falls and bounces by the laws of physics. What physics laws would change because it was previously on a 5 instead of a 3?

First, I'll try to give you the intuition of why it doesn't work like that:

Imagine you have one regular die 1-2-3-4-5-6, and another die with colours instead Blue-Red-Yellow-Green-Orange-Pink.

You roll the first die and obtain 6, that was a 16%.

You then roll the second die. What would that mean to "have a smaller chance to roll a 6 again"? Which colour is "6"? Is that Pink because it was the last of my list? That's completely arbitrary. It doesn't make sense that my first roll with change the probability of obtaining a colour or another, so there is also a 16% chance of obtaining each colour.

Now, if I write some numbers on top of the colours, does that magically change the probability of obtaining the "Pink 6" just because I obtained a 6 before? Surely not.

And if instead I rolled the first die a second time, why would that be any different? It's not like objects have some "magical memory" that remember how often it rolled 6.

Now, here is why you did think it worked like that in the first place:

If you roll the first die, don't look at its result, then roll the second die, then yes, the probability of getting two 6s is quite low: 1/36 = 3%

Similarly, if you roll the first die, don't look at its result, then roll the second die, and then I look at both and say 'I promise you that there is at least one 6', then the probability of a double 6 remains quite low: 1/11 = 9%. What's the difference? Well, I didn't tell you whether the first or the second was a 6, so you have less information than in your example (where you knew that it was the first who was a 6). Just a slight change in information and the probability is different.

And lastly, in nature, there is a lot of things that look random but aren't. For example, there is a 50/50 chance of being night or day, but after enough night the day eventually come, because that's not actually random in the same way a die works.

But in the situation you described:

• You have total knowledge about the first roll. It's a 100% chance of obtaining a 6 because you know the 6 happened. That's a fact that cannot be changed.
• You have zero knowledge about the second roll. It's a 16% chance of obtaining a 6 because the roll is fair and nothing is influencing it.

Let's say you roll 2 dice one at a time. There are 36 possible outcomes. You always had a 1 in 36 chance of rolling one of these combinations. That means that rolling a 6 then a 6 was always 1 in 36.

After you roll the first dice, you can no longer roll the other 5 numbers on the first dice, so as a result the chances of 30 of the combinations coming up is now 0%. The remaining 6 combinations are now 100%, up from 16.6% before the first roll.

The logical fallacy is that rolling a 6 and a 6 was always a 1 in 36 chance. But after you've rolled the first dice, you're not just asking if you're going to roll a 6 again. You're also asking what's the probability that I'm going to roll a double, which should be a lower probability right? The problem is that rolling 2 6's wasn't the only possible double before you rolled the first dice. You always had a 6 in 36 chance of rolling a double (1, 1 or 2, 2 etc). After rolling that first dice, you only have 1 chance to roll a 6 but there's no longer 36 possibilities, there's only 6.

The 1st time you roll a 6 sided die, you have a 1/6 chance to roll a 6.

The 100th time you roll, you still have a 1/6 chance because it's still a 6-sided die.

The 100000000th time you roll, you still have a 1/6 chance to roll a 6 because it's still a 6-sided die.

If you have rolled four 6s in a row, then on the 5th roll you have a 1/6 chance to roll a 6, because it's still a 6-sided die. Completely apart from that, the chance of rolling five 6s in a row is 1/7776, because there are 7775 other possible combinations for those five rolls.

All the results of dice throws are equivalent to each other... if you pay attention and take note of EVERY die number, that is. But if you're focusing on just the number 6, then the other numbers are all considered "not 6"s.

The "not 6"s are 5 times more likely to appear than 6 on a six sided dice. The probabilities are drastically different now.

The chances of a 1 being rolled after a 6 is the same as a 6 being rolled. But the chances of a "not 6" being rolled after a 6 is much greater.

A "13554663" result is just as likely as "66666666" to occur. But if you only note the 6s, then "########" (all "not 6"s) is much more likely than "66666666" to happen because "########" fits with multiple possible results while "66666666" fits with only one possible result.

Dice don't have a memory. They absolutely don't care what has happened already. Every time you roll one, you're starting from scratch, simple as that.

The dice have no way to remember the times you've rolled them before. Since they don't know you rolled a 6, they can't act differently on later rolls to make 6 less likely.

Because they are independent events.

Think of it this way - if you roll a die in Chicago and I roll a die in New York - they are not connected in any way. Agree?

Same is true if you roll the same die repeatedly in the same place. It has no memory.

It is possible to design a system that tries to balance outcomes explicitly so as to “load share” based on some memory but that is not how ordinary randomness works.

As many said: whatever you rolled before, the next roll will always have a 1/6 chance of becoming any specifuc number. There's 6 different outcomes, each is equally likely.

I'd like to add: When figuring out the probability of the sum of multiple rolls, you look at the number of combinations that make that sum.

There's only one way to make a 2 with 2 rolls (1 and 1).

There are two ways of making a total of 3: 1+2 or 2+1

There are three ways of making a total of 4: 1+3, 3+1, 2+2

5 = 1+4, 4+1, 2+3 or 3+2

And so on.

The probability of a certain sum (12 for two sixes) is the number of combinations that give you that sum (1) divided by the total number of combinations there are (6*6), so it would be 1/36.

If you wanted another specific outcome, say the first die must be a 3 and the secomd one must be a 1, that's also 1/36. If the sum just has to be 4 the probability is 3/36 or 1/12.

Dice have no memory. What happened in previous rolls in no way affects the outcome of future rolls. The odds of rolling a six don't change to "make up" for not rolling a six in a while.

If I say I am going to roll a die 4 times, the probability of getting 4 of the same number I picked in a row is 0.077%

Each time I roll the die, the probability of getting the number I picked is 16.67%

The probability of each roll achieving the result you want actually increases as the previous rolls are in your favour.

If you roll 2 of the same in a row, the odds of getting two more is 2.78%, and if you get 3 in a row, the odds of getting the final one is again 16.67%.

I think you're confusing two concepts:

1. The chance of any particular number for a roll of the die once (e.g. 1 in 6)

2. The chance of getting a series of specific results. For instance, the chance of not rolling a 6 if you rolled 30 times is quite low (something like .4% chance of happening).

However, the fact that a sequence of rolls is likely to occur is NOT predictive of any single roll. You know it's weird if you roll 30 times and never hit a 6, but if you roll a 31st time, your odds of any one number is still one in 6.

Don't confuse the probability of getting a sequence with the probability of any single roll. It's a little counter-intuitive for some people.

One thing I'm not seeing is that we tend to look at probabilities being 'fair' - If I roll a die 6 times, and it comes up 1, 2, 3, 4, 5 and 6, that feels 'fair' because each face has an equal probability to roll.

However, if I roll the die 12 times and it comes up with three 1s, three 2s, two 3s, two 4s, two 5s, and no 6s, that feels unfair - and it feels like we should roll two 6s to make it fair and match the probability.

However, probability doesn't really behave that nicely in the small scale. If I roll that die thousands and thousands of times, yes the results will be roughly equal per die face, but that doesn't mean that each set of 6 rolls maintains the large scale probability.

Every roll of the die is an completely independent, right? There's no way that your previous roll can effect your next roll. That means you might go 30 rolls without landing a 6, or you might roll a 6 ten times in a row. But averaged over many many rolls 1/6th of them should be 6s.

If you ask “what is the chance I roll a 6 on this one die roll?” That never changes at 16.7%.

But if you were to ask ‘what if I roll twice and get at least one 6?” That’s 30%

Chance on 3 rolls? 42%

6 rolls? 67%

But if you rolled 5 times and didn’t get a 6 yet, the chance your last roll is a 6 remains 16.7%.

The thing you might be looking for is the Law of large numbers. The theorem states that the average of the results obtained from a large number of trials should be close to the expected value and tends to become closer to the expected value as more trials are performed.

So if you roll the dice 1000 times, it is astronomically unlikely to never roll a 6, or that all 1000 will be 6s. It isn't impossible, but very unlikely. The distribution of how many times a number was rolled should approach 1/6 for each number.