A probability of 0 doesn’t mean impossible. Similar to a probability of 1 doesn’t mean certainly. If a dart hits a point the probability of the dart hitting that point was 0. Yet it did.
No, the probability of a dart hitting a part of a dartboard isn’t zero. The dart’s tip has a non-zero width…
There’s a fallacy here, although I don’t know exactly what it is. If you use an algorithm to select some real number, the probability of knowing that number in advance is zero. But the probability of “picking a number” is not zero: as you note, it happened.
After an event has happened, the “probability” is always one. The “probability” of WWI occurring after the assassination of an Archduke in Sarajevo is…one.
(“Probability” isn’t even a meaningful term regarding past events.)
octopus is speaking of course of the usual mathematical idealization of a continuous dartboard and perfect point dart tips, not of some physical claim. And they (and others who have said the same) are correct: Mathematicians do not standardly use “probability zero” as synonymous with impossible. Impossible events have probability zero, and also some possible events have probability zero.
In exactly the same way, a mathematician will standardly say that a line segment has zero area, but this doesn’t mean there aren’t any points on the line segment. Empty regions have area zero, and also some inhabited regions have area zero. The portion of the Earth inhabited by unicorns has area zero, because there aren’t any unicorns, but also the Equator (idealized precisely, to gossamer thinness) has area zero, though the latter does contain many particular points.
As septimus notes, one might wish to speak differently. Instead of saying that idealized lines have zero area, we might wish (non-standardly) to say that they have positive but infinitesimal area, and so on, and in the same way, we might speak of infinitesimal probabilities. Very well, there’s nothing wrong with this, although it is tricky to make work well with some of the other sorts of things we often like to say. (I will perhaps speak more on this later.)
But, again, the key point is this: “probability zero” is not standardly taken as synonymous with “impossible”; mathematicians do not generally use the language of probability in such a way as that one can expect to determine possibility vs. impossibility simply from knowing the probability of an event, anymore than one could expect to determine an object’s color simply knowing from knowing its length. The only relationship is that impossible events have probability zero, but the reverse is not guaranteed.
I agree, something about “probability” and “infinity” together doesn’t smell right.
Am I thinking through this correctly?
P = 1/(6^n), for n > 0, where P is the probability of the occurrence and n is an integer, representing the number of rolls, for consecutively rolling the same specific number (or rolling a specific pattern).
Because the numerator is fixed and the denominator increases, as n approaches infinity, the limit of P approaches 0.
As n can never become infinite, P will never equal zero. No matter how many rolls you demand, there is always a non-zero probability that you will roll the same consecutive number every single time. Probability isn’t just the analysis of what can be physically done, but what could be done if physical restraints (imperfect die, wear, time, etc.) are ignored or controlled for.
[Aside, if all you are looking for is consecutive rolls of the same number, the equation becomes P = 1/(6^(n-1)). Your first roll will always be consecutive with all other rolls performed so far, and thus P = 1.]
Even with a circlular cross section the probability of that exact area being hit is still probability 0.
Probability doesn’t change just because something happened. The odds remain the same.
If you use a true random system to generate a real number, then the probability that the system will select exactly 0.78 is zero.
It’s also impossible, because it won’t happen. Go ahead and try. You may get real numbers within some neighborhood of that exact number, but you won’t get that exact number. The probability is zero, and it is also impossible.
I’m serious; go ahead and try. (The system has to generate all real numbers within a given finite region. Of course, such a system is impossible, so there’s no actual contradiction here.)
For a dart board of 3 ft[sup]2[/sup] ( = 432 in[sup]2[/sup]) and
a dart tip cross section area of 2 x 10[sup]-5[/sup] in[sup]2[/sup], we have
2 x 10[sup]-5[/sup] / 432 in[sup]2[/sup] = 5 x 10[sup]-8[/sup], or
0.000005% probability … small but not zero.
No. The exact area has an exact center which is a single point. So each cross section of a circle which has a single point for the center has a probability of 0 for that precise circle to be hit.
It’s not impossible. You could say that it’s 0 for any number in the range that could be selected. It will select a number and the probability of that selection will be 0 and it will be possible. Probability 0 and impossible are two distinct concepts.
Isn’t this akin to the Texas Bullseye fallacy?
Anyhow, no one specified how long in between rolls. Roll once, then wait for infinity to roll again…
No. This isn’t retroactive justification. This is how the math is defined/calculated.
It is impossible for the dart to hit any specified number. It’ll hit somewhere, but its location will not be specified. Or at least, not unless you do it retroactively, defining it by the location where the dart hits, but then the probability is exactly 1.
It’s not impossible. Almost surely - Wikipedia And the probability of it hitting where it hit was 0.
Are we asking the probability of a point on the field of a plain? Not sure that works, does it? Probability is the ratio of successful trails divided by the total number of trials. Our successful trial is a one dimensional object, but our total trials are two dimensional objects. Can we perform this division?
I think we have to calculate probability using some definite area as our success, something larger than the infinitesimal area dA.
I just get itchy when we’re mixing numbers of different dimensions, makes my skin crawl.
But in the post I originally quoted, you mentioned someone basically throwing a dart at a finite space but with infinite area. And after throwing the dart, you’d say, “Look where it hit! What are the odds to hit that specific point?!”
You have to choose the spot first for it not to be a Texas Bullseye fallacy.
It’s not a fallacy to say, after a person shoots a dart which lands at a particular point, “The prior probability of the dart landing at that particular point was very low, even though the posterior probability, conditioned on the information that the dart did indeed land at that point, is very high”.
Not at all. The probability doesn’t change depending on you choosing or not choosing. If I roll a 6 sided die and get a 4 it’s a 1/6 chance that that event would occur regardless of any prediction. After the 4 was rolled it’s still a 1/6 chance that a 4 would have been rolled. Probability is simply a ratio and doesn’t depend on prediction. Probability doesn’t change because of what happened.
I flip two coins and get the sequence HT. That’s a 1/4 probability event. It’s a 1/4 probability event regardless of it happening, not happening, happening because it was predicted, or not happening even if predicted. If you construct a tree of all possibilities and look at the ratio between the properties you select in one set and all possibilities in the other set and do some simple division you get your fixed probability.
The Texas Bullseye fallacy is irrelevant. One isn’t making any special claim about what was hit. One is only stating the probability of that particular event. It’s settled math that a real selected from an interval of reals has a probability of 0 for random selection. Yet as many sources state probability of 0 is not equivalent to impossible just as the converse of probability of 1 is not equivalent to certain. Read that wiki article for a short and readable explanation.
Hmmm. A point on a line segment has the same issue. And you aren’t even dividing entities with different units. You are dividing 1 point by an infinite amount of points. You aren’t dividing it by the area you are dividing it by the number of points in the area. And points are 0 dimensional.
So set of 1 point / set of infinite amount of points. Not a set of 1 point or a infinitesimal area divided by the area.
Hate to triple post but i missed something.
It’s a dart with a finite circular cross section at a target of finite space. There are still an infinite amount of possibilities of where the center of that circular cross section is in that finite area. The probability of any particular circle is 0. Yet some circle will exist.
Two complementary sets exist and each has a complementary probability in this case.
Set 1) the one circle that got hit. Almost never should get hit. Probability 0. Did get hit. Not impossible.
Set 2) the infinite number of similar circles that did not get hit. Probability of 1 for 1 of these circles to have been the circle to have been hit. Almost certainly ONE of them should have been hit. None of them were. Not guaranteed.
And this probability doesn’t change just because something happened.
Not trying to be rude but why is probability confounding?