Can you roll a die and get the same number an infinite number of times?

Factual Questions
41

First place, you cannot roll a die infinitely often. That said, you cannot apply your finite intuition to an infinite situation and expect it to work. Pick a positive whole number. Is it less than

10^10^10^10?

Amazing! Almost every positive whole number is larger. In fact, the chances of picking a smaller number is 0.

The whole thing is developed in a theory called measure theory, which can be thought as probability involving infinite sets. But the bottom line is that a set can have measure 0 without being actually empty. The point is that infinity is not an ordinary number and must be treated with care. Famous physicist Alan Guth made a bad error in his book explaining cosmic inflation by claiming that a certain property of a finite universe would also hold for an infinite one. He concluded that Newton (who had an infinite universe in mind) had erred. But Newton was right and Guth wrong.

42

“A number” doesn’t mean anything to a mathematician. What is a number? It’s a mathematical object that adheres to certain axiomatic properties. That object can be a set or sequence or some other thing, and then when we say “real number” we are just using a shorthand to refer to that other thing.

One method of constructing the real numbers is through the Dedekind cut. This essentially says that every real number is defined by two sets of rational numbers; one set of numbers below the real in question and the other set above. Pi, for instance, will be defined something like:
{{3, 31/10, 314/100, …}, {4, 32/10, 315/100, …}}

Add all of the rationals and a new real number gets squeezed out between them.

You can also construct the reals through sequences that look just like the limits we’re talking about. Use {0.9, 0.99, 0.999, …} to define 1.0 and you can no longer really distinguish between the limit of a sequence and the number itself.

Of course, these objects don’t really resemble the everyday notion of numbers. But then, the reals, the rationals, and even the integers only bear a passing resemblance to our intuitive notions. Mathematicians have to do things the hard way if they’re to have a solid foundation, and that means embracing some non-intuitive things.

43

With a physical (fairly weighted) die wouldn’t the act of rolling it change the likelihood of the next roll? If it happened to roll a six 1000 times then it would be less likely to roll a six again due to wear and tear on the die making the six face less able to sustain the die.

44

If you’re getting to that level of pedantry, then a fairly weighted die isn’t physically possible in the first place. Minor imperfections in the production process will produce tiny, tiny, tiny but real differences in the likelihoods of each face coming up.

45

Those matter after 10^10^10^10 rolls … and that’s barely getting started on infinity …

46

No, you can’t. Go ahead and try: it won’t happen.

You won’t even succeed in rolling the same number 1,000 times on a fair die. Not gonna happen. Try and you’ll see.

Simple statement of fact. This is not going to happen.

47

Y’know, on thinking further about it, I think that it really is absolutely impossible to roll the same number every time on an infinite number of throws of a fair die. After all, what’s the definition of a fair die? It’s a die such that, as the number of rolls approaches infinity, the ratio of the number of any two outcomes approaches 1. But if we roll the same number every time, then that ratio is not one, and therefore our die is by definition not a fair one.

In fact, you can make the case that all events of probability zero are completely impossible. You can say “Pick a real number uniformly at random. What was the probability that you would pick that number?”, but “that number” isn’t actually a specification. You can say that the probability of picking pi, or one, or sqrt(2), is zero, but none of those was actually the number that was picked. In fact, you can never even say what the number that was picked was.

48

There’s also the practical contradiction in the OP’s question: you cannot “do” anything an infinite number of times. Infinity cannot be “performed.” No matter how many times you rolled the die, you would still be required to roll it at least one more time.

An “indefinite” number of times would have been better phrasing, or “an arbitrarily large” number of times.

ETA:

I believe this is called “The Law of Large Numbers” and it is not just a speculative idea, but a proven theorem in mathematics. As the number of trials gets large, the number of successes divided by the number of trials approaches the probability of a success. The same thing goes to an average of a large number of die-rolls.

49

Without calling previous answers “wrong” I want to describe a different perspective. (I await with some trepidation to see whether the Board’s mathematicians will boo me down. :wink: )

It is common-sense that the statements
X is impossible; its probability is zero.andX is possible but its probability is infinitesimal.
be distinct, although in each case the probability has Archimedean measure zero.

In mainstream mathematics the two cases both have p(X) = 0. Indeed the very word “infinitesimal” does not appear in mainstream writings except in informal usage. The “Axiom of Archimedes” is a venerable property almost always assumed; it’s even one of the explicit axioms in Hilbert’s geometry.

But it is possible to devise formal systems in which the two X’s in the indented statements above have distinct probabilities. Here is a pdf paper with such a treatment. One can also find papers which introduce non-Archimedean fields to deal with some problems in decision theory or even quantum physics! I hope the Board’s mathematicians will tell us whether such studies have practical value.

50

It’s also possible (we’ve demonstrated it here) to devise a system wherein .999~ <> 1.0 However, it requires a re-definition of “number” in such a way as to allow “infinitesimals” to be manipulable quantities.

It produces a (reasonably) workable mathematics, self-consistent and even vaguely practical.

Darn goofy waste of time…but recreational mathematics is no more a sinful waste than golf, I guess.

51

The probability is zero means both impossibility and extremely low possibility in statistics. Actually in maths we have 0=1-0.9999…(infinite decimal), and this is an equation, not an approximation. Also we have another equation 0=1-1. Both are definitely right in maths, and their left hand sides zero-s mean the same in maths. So if you want to get the same number infinite times, your chances are literally zero, but still possible. Note that here 0=1-0.9999…, not the usual definition, although they are the same in maths. Put it this way, 0=1-0.9999… and 0=1-1 are of the same meaning for maths because there is no distinction when applied to calculation or used for definition. For us they are different, one is possible and another impossible.
It’s easy to understand. We think they are different only because we intuitively suppose these numbers represent something. But maths wouldn’t think so without some premise. For example, 0=1/n with n approaching infinity and 0=0/2, if the situation is to pick one integer from all integers and to pick 1 from {2,3}, then for both us and maths, the two zero-s would be different.

52

" there will always be a number even smaller and smaller and smaller … but never reaching zero."

It does reach zero, that’s what infinity does for you. As an aside, the expression .999~ = 1.0 works because there isn’t a number you can add to .999~ that brings it closer in value to 1.0.

53

I wouldn’t be so sure about that; certainly, nobody’s ever actually done it. After all, almost all of these numbers can’t even be named, and to ‘pick’ a number, we’d have to name it. So we’re really drawing from, at best, a measure-zero subset of the reals. And in fact, we can probably safely disregard all of those numbers we couldn’t write down with all the resources in the universe. So the set of numbers we can pick from is, in fact, finite (and one might suspect that the upper bound on its cardinality given by using all of the universe’s resources is unlikely to be saturated).

54

It seems just a version of a very old type of paradox, like Zeno’s paradoxes. One answer as given is to mathematically formalize what’s meant by infinity. The other, and actually related, theme is to just consider that you will never achieve an infinite number of rolls of the dice.

The example of guessing what real number between 0 and 1 somebody else picked seems ‘scarier’ at first glance, because it seems you could actually achieve the paradoxical point on that one. But the same limit issue is just slightly obscured there. The real numbers between 0 and 1 ‘virtually’ all have an infinite number of non-repeating digits at the end. Any real world experiment will use only a subset of those real numbers, the ones with some infinitely repeating digit at the end.

55

Is it recorded anywhere how many times someone actually has rolled the same number on a (presumably fair) die? Is there a posted record anywhere?

(Googled; no answer.)

Betcha it’s less than twenty.

56

If we’re talking about infinity at all, then we’ve already left the real world. One can take an ultra-intuitionist approach to math where we aren’t allowed to argue about things that could not plausibly happen in the real world, but that leaves a lot of interesting math on the table. If you can’t have infinite rolls, then the odds of rolling the same number each time are just a small but non-zero value.

I don’t think “pick a random real number” is all that different from, say, Cantor’s diagonalization argument where he constructs a real number from a (countably) infinite number of other real numbers. It requires an infinite number of “steps”, but each step operates on a digit with a finite index, so there’s no real difficulty.

57

There are some you tube videos of people trying to roll natural Yahtzees (five dice with the same value).

Maybe you could convince 1 billion people to roll a die about 130,000 times each. At one role every 6 seconds that is about 216 hours. Totally doable over the course of a year or two. Just give up watching “two broke girls” reruns. I am sure other countries have some bad TV to give up. It could be a movement, Probability for Peace.

58

Google mentions 32 consecutive reds at roulette in 1943 (though red is nearly even-money compared with less than 17% for a die roll), but I’m skeptical — one Google hit says the run was in America, another says France. :confused:

On May 3, 2009, Patricia Demauro held the dice for 154 rolls at an Atlantic City casino. Whoever wrote the article didn’t know much about craps: It says she never crapped out (Not necessarily, you keep the dice when you roll craps); it says she rolled no seven until the end (likely wrong, seven is good on the come-out shake.) Still 154 rolls — if we can believe it — is amazing.

Just now I teleported to Monte Carlo and played a full 20 million rounds; I got to roll 70 times or more on only 93 of those rounds. :frowning: I did have one round with 21 consecutive passes — no crapouts — though the total rolls for that round was only 58.

Like many desperate gamblers, I continued beyond the original 20-million round plan. At round 39,797,819 I held the dice for 107 rolls! (though got only 10 passes). Round 105,607,415 was rather pathetic. Despite 105 total rolls, I had only 8 passes and crapped out once — I hope Mrs. Demauro’s round wasn’t anything like that.

After 245 million rounds I tied the record of 107 rolls set on round #39,797,819. Finally at round #385,200,900 I got 126 rolls before passing the dice (14 passes, 0 craps). Twice (on about the 51 millionth and 560 millionth rounds) I got 26 passes with no craps.

After a billion rounds I came home. The 126-roll round and a 112-roll round about half-way through were the only rounds with more than 107 rolls. :o

Based on this experiment I’m somewhat skeptical of Mrs. Demauro’s claim. Perhaps she was rolling improperly and her so-called “rolls” included throws invalidated by the casino’s box-man. (To be fair, the casino claimed the record for longest round by time — 4 hours 18 minutes — not number of rolls, and the 154 rolls might have just been a flawed estimate. Perhaps she was just a slow shooter.)

59

You guys are being nicer than I expected. I was expecting someone to say dude, you’ll die long before you reach infinity.

Good to see that Vi Hart is still making videos. It’s been awhile since I thought to check.

60

Any talk of infinity is speculative at best. Can a die always roll the same number, or a coin always come up heads? Theoretically maybe, but there’s always that next flip that could blow the whole theory. Always. So one can not say for sure. I can’t roll the same number ten times in a row or get heads, let alone an infinite number. So I’d say, for all practical intents and purposes, no.

Infinity is too big a concept for human minds to ponder. And yet we look it right in the face every night when we see the blackness of the sky beyond the stars.

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