I was watching a video by Vi Hart on YouTube and she mentions the chances of rolling the same number on a fair die an infinite number of times is literally zero.
While I get your chances of rolling the same number decreases with each roll I thought it never reached zero. It just infinitely approached zero without ever quite getting there.
I do not know Vi Hart’s qualifications but she seems on the ball with a lot of this stuff. What am I (or she) missing?
EDIT to add: I also get each roll has the same chance as the one before it so rolling a “3” is exactly as likely as the roll before it. Just the chance of me rolling infinite threes approaches zero but never quite gets to zero.
I wonder about this too though I think it has to do with the law of small numbers, maybe its a probability vs. statistical thing. Even though if you flip a coin it could conceivably come up heads forever, it tends to even out to 50% heads, and 50% tails given a large enough sample, but that sample could be very, very large I think.
I would think it would be technically possible to infinitely roll heads forever, but from a statistical standpoint so unlikely as to approach zero, but not actually be zero, someone correct me if I’m wrong.
Is this akin to the 0.9999999~ (to infinity) equals 1 deal? If you get infinitely close to a number then you are that number even if you never “quite” became that number? (always had a problem with that)
Somewhat counter-intuitively, it is possible for an event to happen even when the odds are zero.
Pick a random (real) number between zero and one. This is obviously possible, and yet the odds of having picked that specific number are zero.
Something similar is going on here. If we imagine picking a number in base-6, then rolling a die with the same number infinite times is like picking 0.111…, 0.222…, etc. The odds are zero of picking one of those, just like any other number, and yet it’s not exactly impossible.
ETA: Yes, in a way this is related to the 0.999… = 1 thing. 0.9 + 0.09 + 0.009 + … doesn’t just approach 1.0, it is 1.0. Same deal here. Infinity works in mysterious ways.
You say my chances of doing “X” is possible yet zero at the same time.
I can’t reconcile that in my head. I have a literal 0% chance of something happening yet it can actually happen (albeit an admittedly, ridiculously remote chance).
That’s the trouble with infinity. It is, by definition, larger than every other number in your number system. And so if the odds of something go like 1/infinity or 0.5[sup]infinity[/sup], then it must come to a number that’s smaller than every other number in the system–which for the real numbers is strictly zero.
It’s possible to use enhanced number systems that avoid this problem; they add “infinitesimals”, which are numbers that are closer to zero than any real number but aren’t zero. They behave a bit like if you added 1/infinity to your system (but with some features to ensure they work consistently).
But leaving that aside, all we can say is that the odds are zero. Not very satisfying, but true.
Strangelove: why are the odds of picking a specific number between 0 and 1 zero? Is it because there are an infinite number of numbers between any two numbers and since all numbers after the decimal are infinite then there is no end to any given number?
You need to get the idea of infinite sorted. Axiom of choice comes next. In both cases intuitive notions don’t work well. Infinity is not just a very very big number. It does not obey the same rules, and glib use of it in describing things always causes problems. The OP’s cite could be restated as saying that you will eventually throw a different number. No difference. But it avoids the difficulty with handling arithmetic and infinity.
Is there some reason if I roll a fair die that I cannot roll an infinite series of the same number?
If each roll has a 1:6 chance for a given number and each roll is independent of the roll before it why can’t I, in principle, roll an infinite number of 3’s?
This is what happens when people try to talk about infinity without mathematical rigor. As the number of rolls of the die goes to infinity, the probability that you roll the same number each time goes to zero. You can formalize this as a limit. Without formalizing “infinity” as a limit or in some other way, you’re unlikely to learn anything useful (beyond that infinities are unintuitive).
And yes, non-impossible things can happen with probability zero. The usual phrase is “almost never” or “almost surely”.
Probably a real mathematician can answer this better, but I’ll give it a shot.
Let’s suppose we pick our number with a 10-sided die, and we want to match some pre-determined number, like 0.222… or pi.
You’ll agree, I think, that the odds of rolling the first digit correctly are 0.1. And then the odds of getting the first two digits right are 0.01, and so on.
You’ll also agree, I hope, that the odds of getting N+1 digits correct are lower than getting N correct.
So suppose I just ask you a series of questions:
Are the odds (of rolling infinite digits in a row) less than 0.1?
You answer yes.
Are they less than 0.01?
Yes.
Than 0.001?
Yes
…and so on.
So we ask: what (positive) number is less than any other number that I could name? You have an infinite number of digits; no matter what non-zero number I name, no matter how many zeros I put in front of the 1, the odds are less than that.
There’s only one number that fits the description, and that’s zero. Note that 0.000…001 is not a real number–that’s just not allowed when constructing the reals. I can put a million, or a trillion, or a googol zeroes before the 1, but I can’t just stuff infinity zeroes there. But if I say that there are google zeroes, then you just reply that you’re looking at googol+1 rolls and the odds are still less than the number I named. The answer is zero because it can’t be any other number.
I get there are an infinite number of numbers between 0 and 1.
But the chances I pick your number can’t be zero. It can be incredibly unlikely I guess your number but that chance can never actually be zero. Yet the woman in the OP is saying it is literally zero. Not close to zero but actually, literally zero.
Since infinity is not a number how can you get zero from it?
If not zero, then what other number? No other real number fits. Your choices are either to introduce a new class of numbers, or accept that “probability zero” doesn’t mean “can’t happen”. Both ways could be made rigorous, but the second one is easier.
Maybe my problem is she is using a concrete example then saying it is infinite.
For example:
I can roll a die.
I can roll it again…
And again…
And again…
And again…
I can keep doing that but I cannot do it infinite times. No matter how long I roll the die it will be finite. And since I can never, ever do it infinite times it MUST be possible to roll the same number endlessly but not infinitely.