You need a formal definition of “infinity”, because your intuition will almost surely fail you. Adding “endlessly” into the mix, also without a formal definition, is probably a step in the wrong direction. Have you studied any calculus?
It is possible to roll the same face of the die any number of times. Using the definitions that most mathematicians find to be most useful, as the number of rolls goes to infinity, that remains possible, but the probability goes to zero.
First off, one can’t roll a die an infinite numbers of times … every time you roll, you add 1 to the count … you will always be able to add 1 to the count no matter how high the count goes … there is always a number 1 higher. No matter how high, there is a non-zero probability that you can roll the same number that many times.
This probability gets smaller with each increase in the number of rolls, but again it doesn’t matter how many times, or how small the probability gets, there will always be a number even smaller and smaller and smaller … but never reaching zero.
However, we can say that as the number of rolls gets closer and closer to infinity, the probability will get closer and closer to zero. Since we cannot roll the die an infinite number of times, we can never reach zero probability.
“Closer and closer” isn’t very a mathemagical sounding concept, so we actually say “the limit of the probability as the number of rolls approaches infinity is zero”, or even more fancy-pants “dP/dR = 0” where P = Probabiity, dP is the differential of Probability, R = Number of rolls and dR is the differential of the number of rolls.
Remember, we can never roll the die an infinite amount of times …
My recollection from Stats and Probabilty class says that rolling the same number an infinite number of times would be zero (or at least approach zero, in math terms). However, if you choose a number of rolls, no matter how high that number is, we could put odds on that.
Also, just to mention it, keep in mind that, for example, the odds of rolling 4, 4, 4, 4, 4, is exactly the same as rolling 2,5,2,3,1. Rolling the same number over and over is no different than rolling any other predetermined sequence of numbers, it’s just that a repeating die roll catches your attention. Just like you have the same chance of flipping heads 10 times in a row as you do of flipping any other pattern that you pick out ahead of time.
If we roll the die R times, and P is the probability that we get the same number each time, then dP/dR is certainly not zero everywhere. dP/dR does go to zero as R goes to infinity, and P does go to zero as R goes to infinity, but the first does not imply the second.
I don’t know much about advanced math, but it seems to me that everybody is missing the point.
If you roll the die an infinite number of times, there are still only 6 possible outcomes. Therefore, it logically follows that each of those outcomes will occur an infinite number of times.
Yes. Exactly what I was going to say.
The answer is, “Yes, if you roll it an infinite number of times.”
Bolding mine
We get the point, but you ask an absurdity … infinity is not a counting number, it’s not a number at all … it’s a quality. Might as well ask “roll the die a red number of times” or “… a Wall Street Journal number of times”.
You’re premise is wrong, we cannot roll the die infinity number of times.
Since we are talking about a physical object - The die will eventually be worn away, so it will not even reach ‘astronomical’ numbers of throws.
Since the odds of rolling any given number are one to five, then no it is five times more likely that you won’t get the same number ten times or a hundred times in a row, much less a much larger concept like a trillion trillion or infinity. There’s the theoretical number (especially if the die is weighted), but for all practical intents and purposes, no.
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Ok sure but the OP has a point, there should be a difference between, the probability approaches a limit of zero" and zero. The second means it can’t actually happen, the first means it can happen but with a infinitesimal chance, this strikes me as an important distinction.
Just like with the .999… threads I want to point out that the phrase “What number represents the chances of rolling the same number an infinite numbers of times?” doesn’t make sense. The “odds” are not an actual number. What you can say is that the limit is 0. But you cannot actually say it is 0.
An infinite sequence* and the limit of an infinite sequence are two different entities with different types. One is a sequence, the other is a number. It would be like equating {0} and 0. One is a set, the other is a number.
A die has six sides. A chance of rolling the same side twice in a row is better than rolling 3 or more times. The more rolls the less chance so she could be more right than not.
The usual mathematical interpretation of the phrase “the odds of getting ____ when rolling a die an infinite number of times” is “the limit, as n increases without bound, of getting ____ when rolling a die n times”. This limit is very easy to calculate using basic calculus, and is equal to 0.
If you don’t know basic calculus, then we can start with a primer on that. Alternately, if you have some other interpretation of that phrase in mind, let us know, and we’ll work from there.
Not from a mathematical point of view, but I don’t find it counter intuitive at all.
It’s counter intuitive only if you equate “infinite number of times” with “a very very very large number of times”.
Since you will never ever stop rolling the dice until you get a different number, you will eventually roll something else than 3. You might have an extremely small chance of rolling only 3s for ten billions of billions of billions of years, but you have no chance of getting only 3s if you roll it forever.
I don’t really see how it’s not intuitive that the likelyhood of rolling only 3s is higher than zero only if you limit the number of throws.
Mathematics has certain convenient concepts that never happen in real life. Infinity, and imaginary numbers, are both examples.
Zero divided by 6 is still zero. In exactly the same way, infinity divided by 6 is still infinity.
Nobody is talking about getting the same result in a row. We all know that that’s not going to happen. We’re talking about total number of outcomes for each number on the die.
In case this hasn’t already been answered satisfactorily:
It’s true that your chances of rolling the same number never reach zero. This is for precisely the same reason that the number of times you roll can never reach infinity. You have to be consistent with your “never reaching.”
I own one with one side (its round). My odds of rolling that an infinite number of times and always getting the same number I guess would be higher than zero.
This is wrong, and it seems to be what’s confusing some people. Probability zero does not mean that something can’t happen, nor that it’s impossible. That’s not how probability works.
If there are infinitely many different things that could happen, they can’t all have nonzero probability that they would happen. (A mathematician would make this more precise by talking about the axioms that probability assignments have to satisfy and noting that, in an infinite sample space, there’s no way of satisfying those axioms unless the probability of at least some events in that infinite sample space is 0.)
Here’s one way of looking at it:
Consider all possible infinitely long strings of numbers for which each number in the string is either 1, 2, 3, 4, 5, or 6.
“Rolling the die an infinite number of times” means randomly selecting one of these infinitely long strings of numbers.
Among this set of infinitely long strings, there are certainly some that do not contain all six numbers, like for example “111111111…” and “1232123212321232…” If you’re randomly selecting one out of this set of infinitely long stings, there’s no reason you couldn’t get one of these, so it is possible that one or more of the six die rolls would never come up at all, let alone an infinite number of times.
I suspect that the probability of this happening would be 0; but again, that’s not the same as saying that it can’t happen.
Right. And the chance of picking any string, even ones that look “more random” is still zero. Rolling a 6 infinitely many times in a row has the same chance as rolling “1245324233326512…”