Except that is would take over a year to actually buy 14 m tickets (assuming 1 ticket purchase per second for 8 hours a day).
13983816 seconds (assuming he can buy 1 ticket per second) 3884.39 Hours 485.55 Days (assuming 8 hours per day)
I agree that this has become a bit exotic, and probably irrelevant for the analysis of realistic games. Most if not all games machines will have a finite space of outcomes, which you are trying to predict. In those cases, a probability of '1' means certainty.
The first infinite space we encounter in school is the set of integer numbers: 1,2,3, 4,5,... and so on, with no upper limit. So if I wrote down a number, and asked you to guess it, the probability that you would get it right is zero, but it's not a certainty that you wouldn't get right.
But you can't buy random tickets. You just buy tickets. What numbers you put on them is up to you.
"johannes" wrote
Just because you wrote down an integer, doesn't mean that there was a chance that it could have been *any* of the integers. You would have been writing down a number from the finite set of numbers that it is possible to write down on a piece of paper. For instance, there are an infinite number of integers that you simply wouldn't be able to fit onto even the largest piece of paper, even if you used notation such as "1 x 10^999".
"johannes" wrote
That's because the probability of getting it right would be greater than zero. I wouldn't have to even consider an awful lot of the integers, because there'd be no way that you could write them down!
True. But it's then interesting to ask: If there is 'only' 63% chance, then where does the money goes? You would expect at least a return of the winner's pot when buying that many tickets which matches all the possibilities.
The 63% is the probability of 'at least one' win, i.e. the opposite of not losing. Hence the answer is that you do get all of the winning pot (in a statistical sense), some of it by multiple wins.
Now you're nitpicking ;-) Of course I'm in possession of infinite paper and I'm able to write numbers down in zero amount of time.
"johannes" wrote
No, I'm just being realistic - there's no point in considering "scenarios" that don't exist!
"johannes" wrote
If *you* can do that, then *I'm* telepathic and therefore the probability of me getting it right *is* one and it *is* certain I'd be right!
But now, back in the real world... Can you devise *any* *real* experiment where it is possible to know "for sure" that a particular outcome has happened, but which actually had a probability of happening of zero? No?
[..]
The probability of something, which is not imaginable, must be zero by definition, yet not impossible.
"johannes" wrote
By which, I assume you mean that it is not even possible for anyone to imagine this "something"...
"johannes" wrote
Eh? That which is "not imaginable", *must* be impossible...
Because: if something is "not impossible", then it is possible. If it is "possible", then it is certainly "imaginable". Hence somthing "which is not imaginable" *cannot* be "not impossible".
Now - would you care to answer my previous question? [See top of this post.]
Something might not be imaginable at a certain time, yet happen at a later time. Of course, when it happens, it makes a transition into something imaginable, but you're not allowed to revise probabilities since they apply to the situation at the time.
"johannes" wrote
No - it was "imaginable" all along! If "something" is capable of happening at some time in the future, then it is capable of being "imagined" now.
"johannes" wrote
That's OK - the probability of it happening didn't change, it remained greater than zero throughout.
Would you care to suggest anything at all that has actually happened (at any time) in the past, which you think that it would have been impossible to have been imagined beforehand?
No, the probability was zero before the event, then changed to greater than zero in the light of experience, as in Bayes Theorem.
Probably many things, but I don't want to give you ammunition ;-)
Big Bang?
"johannes" wrote
Tee hee! Which of the above (which are meant to apply to the same thing), do you really mean? ;-)
"johannes" wrote
I respectfully suggest that *anything* that has & can happen *is* capable of being imagined, even before it has ever happened. I await evidence to the contrary... ;-)
"johannes" wrote
That was certainly "capable of being imagined". We can now only "imagine" that it actually happened anyway
- we have no direct evidence that it definitely *did*!
I guess your next question will be "who imagined it?" ! Well, (obviously) the answer is "God and/or the intelligent life forms existing between the previous 'Big Bang' and 'Big Crunch'."
But anyway, you don't need a "conciousness capable of thought" to exist, for something to be "capable of being imagined". You just need it to be something that, if you put a "conciousness capable of thought" there at that time, it would be able to imagine it. [In other words, the lack of the "conciousness capable of thought" does *not* make the probability zero for the thing "capable of being imagined"!]
Only with hindsight, and that doesn't count, e.g. Bayes again.
On the contrary, the probability must capable of being assessed before the event. Otherwise it's 'cheating', like looking up the solutions, or betting when you know the outcome.
"johannes" wrote
NO! - Just as we could sit imagining the next 'Big Bang' to happen (which would be after a 'Big Crunch'!), even
*before* it has happened (ie, not with hindsight), so could lifeforms who lived before the last 'Big Bang' (and before the previous 'Big Crunch') sit around imagining our own 'Big Bang' being about to happen (even before it did)."johannes" wrote
Bayes has nothing to do with it. Would you care to elaborate how you think Bayes is relevant?
"johannes" wrote
Are you seriously trying to suggest that, before the mathematics of probability had been invented, and so no-one could "assess the probability" of anything, that the probability of everything was zero?! Don't be silly...
"johannes" wrote
The toss of the first ever coin made, (may have) had a probability of 0.5 for "heads" and 0.5 for "tails",
**even if** no-one was able to assess it beforehand!
Not at all. The two things are very much interconnected.
The probability of heads coming up on a coin toss is 50%. Which means that *on average* over x coin tosses there will be x/2 heads that come up.
Which doen not mean that it might not come up at all over a finite number of tosses.
Yes, but only *per game*. If you were to do it for every game, the odds of winning the same number of times as the number of games you play increses, until is approaches certainty.
Which is what I tried to convey by the words, "on average".
In response to what Cynic posted in news: snipped-for-privacy@4ax.com:
It means no such thing. It means that the more coins you toss, the more likely it becomes that the proportion of heads will approach 50%.
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