Shade is it ever possible to actually pick a random number (or any number) between 0 and 1. The number would have to be defined to infinite precision.
Can we say that “Chosing digits randomly to make an infinite string” is even theoretically possible. Are we really justified in saying that any given string can be infinitely long, and can be compared to another infinitely long string of digits?
Thinking again of an infinite Universe, is the state of that Universe in its entirity, one of an infinite set of zero probability states it may have been, or is it a single state of probability = 1 ?
That was my parsing of the statement that “The Lebesgue measure of any point is zero, but of the unit interval is one,” which I think is how we were taught to interpet it. I don’t know of any system for actually providing a random number.
But I think that mathematicians define probability in terms of measures , so from their point of view this should be as valid as picking a number from 1,2,3,4,5,6.
But I’m not sure I completely understand your post; am I getting warmer?
Bippy is talking about physically picking a real number between in [0, 1]. Sure, that’s impossible because of the limits of finite precision. But we can still talk about it, and that’s what we’re doing.
Only if probability 0 is equivalent to never happening. As has been mentioned before, that’s a bad assumption.
Unfortunately, explaining this in any detail will get you into some rather advanced math very quickly. I can’t think of a good introduction that doesn’t assume the equivalent of a bachelor’s in math.
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a) is true. b) is false–see the counterexample provided by Jabba
The conclusion of b) is what people are trying to prove about pi.
If you flip a coin an infinite amount of times, then there is a small probability that it will come up heads every time, and a extremely large probability that tails will eventually come up, right? However, given an infinite amount of time, all probabilities will eventually come up. Dores this mean that there is no probability of a coin coming up heads every time in an infinite number of flips, or does it mean that the two contradictory possibilities are both possible?
Well, yeah, but “talking about it” and “rigourously defining it” are two different things. We can talk about the properties of the largest prime, but that doesn’t mean it exists.
I think that the problem here is that we trying to assign a real number to the probability, but it’s not really a real number. For one thing, this space isn’t Hausdorff: given an infinite number of die rolls, getting at least one 6 and getting any sequence at all are both assigned the probability 1, but they are really two different numbers, even though they lack disjoint neighborhoods.
Consider the following pseudo-program and an associated proposition:
Start off by picking any number P in the interval 0<P<1.
Randomly generate a number A in the interval 0<A<1.
If A>P, go back to step 2.
Otherwise, stop.
Proposition: The above program, if run, will eventually terminate.
This program is analogous to looking for a particular event in an infinite universe. If the event has a nonzero probability P of occurring at any given place, we start by checking somewhere and seeing if it’s happening right there. If not, we check somewhere else (let’s say we use a space-filling pattern such that every location will eventually be checked). The proposition is equivalent to the statement that this event is guaranteed to eventually happen somewhere.
I suspect that using only the traditional axioms of probability, this proposition is undecidable in the formal sense—it cannot be proven to be either true or false.
To extend the system to answer questions of this nature, one would need to add an additional axiom equivalent to either this proposition or its negation. I can imagine that someone might feel, intuitively, either that this proposition is obviously true or that it is obviously false.
I personally lean towards the idea that this proposition is neither true nor false; but then again, I don’t believe the Principle of the Excluded Middle ought to apply to the real world, and I suspect mainstream logicians would disagree.
First up, it is important not to think of this 4 thingy as a number, but as a process. Hence when we ask “what is the probability P of an event given an infinite sample size N?” we answer not by saying P=1 if N=4 but that P tends to (6) 1 as N64.
4 is one of those stark reminders that, as Einstein put it,
Whoo too much maths for me already… I understand a little about probability (enough to read this thread without going ga-ga); I think if I was to formulate the OP in a more formal way i’d say “given an infinite universe, an event with an extremely low probability is no more likely to happen than in a finite universe.”. I think this is because in an infinite universe for any event E with a probability ~0 there is always an event G with a slightly higher probability. No?
We can also rigorously define the properties of the largest prime. What’s your point?
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wtf? We have a function from the powerset of infinite sequences to [0, 1] that satisfies the Kolmogorov axioms, and that’s all that anyone requires for a probability distribution.
We can also rigorously define the properties of the largest prime. What’s your point?
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wtf? We have a function from the powerset of infinite sequences to [0, 1] that satisfies the Kolmogorov axioms, and that’s all that anyone requires for a probability distribution.