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If I have an "experiment", the results of which can be classified clearly into "outcomes" (like rolling a die), then I can make a concrete and verifiable empirical claim that "if you repeat this experiment many times, the frequency of this outcome will tend towards this number". This is what I mean when I say that the probability of something is such and such a number.

But then what is the meaning of the phrase "there is a 25% chance of rain tomorrow"? If I repeat the experiment of it being tomorrow many many times... what? It just doesn't make any sense, the definition of probability given above just doesn't apply.

To take an even more concrete example, let's say someone presents you a box containing one million black balls and one white. You get to take out one ball (and only one). If it's white, they give you a thousand dollars. Otherwise, you give them a thousand dollars. They're very stubborn and won't let you refuse to play unless you can give a satisfying philosophical explanation as to why you don't want to. What could you say?

Is there a name for this "paradox"? Is there a solution to it?

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There are several different interpretations of probability and not all of them are are problematic in that regard. Your approach to take the frequency of an outcome as the probability is but one of the proposed concepts.

Others take the probability as the result of a property of a system. If you examine your die, you will notice, that it is perfectly symmetrical in regards of it's geometry, mass distribution, friction etc. So you know it is equally likely for the die to land on each side. If you assume Kolmogorov's axioms, you know that the sum of all probabilities must be one. Now if every side has the same chance to land top and there are six sides, you can simply divide 1 by 6 and you have your probability for every single side. In this interpretation the experiment does not even have to run once let alone several times. You know it, by knowing the die. The same case could be made for the weather: You know the properties of the sun, our atmosphere, heat absorption of land masses etc. It is like the die only way more complex. "The probability of rain tomorrow" is therefor just a result of the properties of the system.

Another approach interprets probability as an expression of our certainty that something will happen. It is no property of the die, it is a property of us - of the amount of knowledge we have about something. If you look at a coin you might assign the probability of it landing tails by 0.5. If you would know more about it, you might come up with a different probability. Assume you learn, that the coin will be flipped by a master-flipper who can flip a coin with such precision that it almost always lands the way he wants. Now you still will not know, what side it will land on: You don't know what the master-flipper aims for. So you will still be forced to go for 50:50. Now he tells you, he aims for tails. Do you believe him? How certain are you, he told you the truth? But now you can move away from 50:50. Say you are almost certain he said the truth and so you go for 20:80. It's still the same coin and the same guy, only the amount of data you have has changed. Probability in this model is just about how certain you are. Applied to the weather-problem: 25% says: "After thoroughly going through everything I know about the weather and the laws of thermodynamics and so on, I am almost certain it will not rain tomorrow. But better bring an umbrella - just in case!"

  • Your very nice post boils down to the last phrase - "just in case!". The fact that something happens only once basically leverages numerical approach to knowledge about the system vs unknown "just in case". Speaking in science terms probability is useless if you CAN take umbrella without big resource demands. Probability of single event makes sense if it is difficult to take umbrella and u should reason about if you should do it or not. – Asphir Dom Oct 21 '14 at 11:39
  • @AsphirDom A chance of 25% is still somewhat high. But I didn't recommend taking a raincoat, gumboots and a rubber raft. Single-event-probability makes also sense if you want to weigh risk against cost. And what's the cost of a umbrella weighted against a 25% chance of getting wet? I don't know! That's a matter of personal preferences I guess. But I feel I didn't get your objection... – Einer Oct 21 '14 at 11:44
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    I did not object :) i complemented why we are interested in probability of single even at all. It has some economical impact. "Should we build a road here based on the current levels and trends in immigration?" Stuff like that. – Asphir Dom Oct 21 '14 at 11:49
  • Whether probability is or isn't a property of us or of the system has no bearing on how to interpret "probability of rain tomorrow is 25%". Kolmogorov's axioms assume existence of a sample space, and therefore of multiple realizations, for such interpretation. It does not apply to unique events. And some psychological experiments show that subjective "degrees of certainty" do not obey addition and multiplication rules of probability. What weather forecasters most likely mean is that they ran their model multiple times and got rain in 25% of them. – Conifold Oct 21 '14 at 21:08
  • Really, your first case is a special case of the second. In the language of Bayesian probability, the first one just implicitly uses the uniform prior. In the second case, you make explicit your prior. – James Kingsbery Oct 21 '14 at 21:29
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When the result of an experiment is a probability, the experiment itself generally involves multiple samples, so your repeating the experiment is not the recreation of a one-time event. If I had a sample size of several hundred, I can confidently say something about the several-hundred-and-first. Even when there are only seven or eight readings taken, you are talking about a probability based on accumulated data, which is warranted.

The case of meteorology here, is a red herring. These predictions are not based on an experiment, they are based on a model. "A 25% chance of rain" means that of the alternative simulations that the model generates, on average 25% of the land in your service area gets (simulated) rain. The model itself is improved continually to better simulate the environment. But there is never a decisive experiment where the model is found to either succeed or fail. Experiments in such a modelling science are of the form 'This tweak improves alignment with the observations when the altered model is run over previous time periods, that one does not.'

So the question does not apply in principle. At the point of application, either running the model over a range of inputs is taking multiple samples of tomorrow, or there is no experiment involved at all.

For me, your other example also has little to do with probability per se, but with the fact money is a bad proxy for real value. My reasons for not taking part in the game with the balls is just that 1) the utility of money (or anything else) is not linear, and 2) safety has a value of its own. I prefer safety over windfall gains, primarily because my life is poorly engineered and the cost of $1000 debt is much higher for me in emotional terms than the added utility of $1000 extra income.

Can either of these examples be tweaked to really ask what you are after?

What you may be after is the infamous 'ceteris paribus' assumption that all science makes. We have to assume observed probabilities behave something like mathematical probabilities unless there is a cause for them to deviate.

This is always questionable in principle -- we do know that some things really are just more random than that. (The direction things radiate from an isolated nucleus, for instance, is just not going to converge.) But it is particularly unlikely in practice, because we never know what might constitute a cause. Over time the ability to repeat our experiments is meant to ferret out the unnoticed causes. But that means all predictions, always, are potentially missing required premises.

Accepting science as a model just involves some articles of faith.

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The frequency theory of probability states that the probability is the relative frequency of an event in some suitably specified class of events. As you noted, this has the defect that it doesn't allow you to say anything about individual cases.

One commonly touted alternative is that probability represents our state of knowledge about some event, but this is a non-starter. If you're ignorant you don't have precise numerical knowledge about what's going to happen, and yet this is what probability delivers when it works.

A better approach works as follows. There is some law of physics that explains some particular measure on the set of possible outcomes that respects the calculus of probability. And if that measure applies to single cases then you have a probability for single cases. It may then follow that for many experiments the probability that the relative frequency of the results approximately matches the probability is high. See these papers

http://arxiv.org/abs/quantph/9906015

http://arxiv.org/abs/0906.2718v1

for an example of explaining a particular measure on the set of allowed outcomes. See also "The Beginning of Infinity" by David Deutsch.

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