Wikipedia:Reference desk/Archives/Mathematics/2011 November 21

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November 21

Probability Zero and One

My friend asked me a rather interesting question and I am curious to think what others here think. Can an event with probability zero ever happen? Can an event with probability ever not happen at all? It is related to the whole discussion of moving away from religion/spirituality/fate/destiny and accepting the role of randomness in our lives (not trying to drag religion here on this desk but just providing context). I have thought about this question but not settled on an answer yet (if there is an objective one). I am thinking of probability zero being zero except on a set of measure zero so an zero-probability event is possible. But if it is possible then the probability isn't zero. HELP! Thank you. - Looking for Wisdom and Insight! (talk) 06:27, 21 November 2011 (UTC)

See Almost never -- Widener (talk) 07:24, 21 November 2011 (UTC)

You say if it is possible the the probability isn't zero. Who sez?

I mean, don't tell me literally who sez. Probably Laplace sez. Probably Fermat sez. But they were working in very simple models — finite atomic Boolean algebras where all the atoms have the same probability. First die comes up 3, second die comes up 2; that's an atom, and it has the same probability as, First die comes up 6, second die comes up 1.

In those cases, it's true that all possible events have nonzero probability. But why should it be true in general? Can you elucidate the intuition that is breaking for you? Or are you just being held back by what was taught in elementary math classes, where the models are like the ones Laplace and Fermat used? --Trovatore (talk) 08:08, 21 November 2011 (UTC)

Under the frequentist definition of probability, a probability is defined as $P(x)=\lim _{n_{t}\rightarrow \infty }{\frac {n_{x}}{n_{t}}}$ where $n_{t}$ is the total number of trials and $n_{x}$ is the number of trials where the event $x$ occurred. It is possible for this limit to be zero even if $n_{x}$ is nonzero for some $n_{t}$. Widener (talk) 08:52, 21 November 2011 (UTC)

Well, honestly, that's not much of a definition (which is partly why I don't think very highly of frequentism). It's only with probability 1 that that limit equals the desired value (or indeed even exists), which makes the whole thing sort of circular. Also it only works if the trials are independent, and in that case, for any finite $n_{t}$, if the probability per trial is zero, then the probability that $n_{x}$ is anything but zero, is itself zero. Sorry for the convoluted sentence. --Trovatore (talk) 09:52, 21 November 2011 (UTC)

If you pick a number between 1 and 2 at random it will start say 1.3, 1/10 of them start that way, then perhaps 1.36, 1/100 start that way. The digits are all random so getting one that starts 1.367842 has a 1/1000000 chance. Going on and on like that to infinity leads to a probability that is less than any positive number you care to mention. A departed quantity in Berkeley's terms. And every single number between 1 and 2 would have that same chance of being picked and yet there is a probability of 1 that one of them will be picked. Measure theory is how people cope with the problem. Dmcq (talk) 11:31, 21 November 2011 (UTC)

So can an event with zero probability ever happen? - Looking for Wisdom and Insight! (talk) 07:27, 22 November 2011 (UTC)

The answer is yes. Widener (talk) 07:31, 22 November 2011 (UTC)

Another answer is no. The event that a random variable assumes a specific real value will never happen, because it takes forever to identify a real number precisely, as it has an infinite number of digits. A random real number is a useful theoretical concept, but it is not a practical concept. Bo Jacoby (talk) 13:41, 22 November 2011 (UTC).

Another answer is indeed "no". However, that answer is incorrect. --Trovatore (talk) 17:48, 22 November 2011 (UTC)

Not if you take a second for the first digit, a half a second for the next digit, a quarter for the next etc. Anyway I can wait. Dmcq (talk) 14:42, 22 November 2011 (UTC)

p.s. you might like Conformal Cyclic Cosmology where Roger Penrose suggests after an infinite time another universe may come into being. So even infinite time may not be all of time. Dmcq (talk) 14:46, 22 November 2011 (UTC)

Does an event, that happens after an infinite time, ever happen? Bo Jacoby (talk) 17:02, 22 November 2011 (UTC).

According to that it may already have happened an infinite number of times. Dmcq (talk) 17:20, 22 November 2011 (UTC)

A probability space is supposed to model experiments or measurements, but no experiment identifies a real number precisely. An event with probability zero cannot be identified, and so it does not happen. Bo Jacoby (talk) 09:47, 23 November 2011 (UTC).

Well that's more a science question considering humans as the measure of the world. A bit like the extreme formalists here who think analysis should only consider constructible numbers. Dmcq (talk) 10:26, 23 November 2011 (UTC)

Not really. It is merely a question of the meaning of the two words: ever and happen. Ever means within finite time, and an outcome happens when it is identified. A probability zero outcome is not identified within finite time, and so it does not ever happen. Bo Jacoby (talk) 08:22, 24 November 2011 (UTC).

Seems strange to say that something that may have already happened an infinite number of times never happens. Dmcq (talk) 11:09, 24 November 2011 (UTC)

Yes, it seems strange, but no probability zero event has already happened an infinite number of times. Bo Jacoby (talk) 11:50, 24 November 2011 (UTC).

That's not clear at all. If spacetime is correctly described by the real numbers, then presumably probability-zero events happen all the time, for example an electron being in one particular location out of the continuous cloud of points it could be at. You seem to be taking the position that something doesn't happen unless it's precisely characterized and observed, but that's just wrong; it has echoes of logical positivism, which is also wrong. --Trovatore (talk) 21:49, 25 November 2011 (UTC)

According to the uncertainty principle no particle defines a position with absolute precision, which is necessary for it to specify a probability-zero event. Bo Jacoby (talk) 23:38, 25 November 2011 (UTC).

parabola or cylinder

in my intermediate i learnt that y² = 4ax is a parabola,now i cant digest that the same thing is a parabolic cylinder. please,make me understand with necessary eqations if needed.59.165.108.89 (talk) 12:15, 21 November 2011 (UTC)

I'm not going to force you to understand, I prefer the softly softly approach ;-) If you draw a parabola on the ground and then start building upwards by placing bricks along it you get part of a parabolic cylinder. It is simply that equation in 3D where z can be anything so making a sheet with a cross section that is a parabola. A cylinder is used here as a general term for a 2Dshape drawn out into the third dimension. Dmcq (talk) 13:06, 21 November 2011 (UTC)

(ec) Technically the graph of that thing is a parabola (or parabolic cylinder). The equation itself is just an equation. When you want to figure out what the graph is, first you need to decide what variables are in play, and in particular how many dimensions you want the graph to live in. In this case you can decide that you're only considering 2 variables x and y, in which case you graph it in 2 dimensions and you get a parabola. But you can also imagine that you have 3 variables x, y, and z, where z just happens to be absent from your equation. In that case you graph it in 3 dimensions (one for each variable), and you get the parabolic cylinder. (By the way, these are not the only choices- you can imagine any number of additional variables and get graphs that live in higher dimensions.) Staecker (talk) 13:09, 21 November 2011 (UTC)

so its a hyper parabolic cylinder? or something like that? — Preceding unsigned comment added by 109.156.115.144 (talk) 13:15, 21 November 2011 (UTC) it means in 3d a parabola acquires a cylindrical form like a circle becomes a sphere. then every thing can be studied about a cylinder just including one more variable likewise we do in a circle for studying a sphere.nd thanx 4 reply,i think i got what u said.59.165.108.89 (talk) 12:16, 22 November 2011 (UTC)

No, the equation y = x² gives a parabola in the xy-plane, and a parabolic cylinder in the xyz-space, just like x² + y² = 1 give a circle in the xy-plane and a cylinder in the xyz-space. — Fly by Night (talk) 22:10, 23 November 2011 (UTC)

then why z is absent 4m the eqation of cylinder if its in 3d nd if it comes then its a sphere59.165.108.89 (talk) 13:20, 24 November 2011 (UTC)