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Say I have a random event $E$ with probability $p$. There is a natural interpretation in terms of $E$ for the probability $p^2$: it's the probability that $E$ occurs twice if I perform two independent trials.

Is there a similar interpretation for the probability $\sqrt{p}$? More generally, given $x \in ]0, 1[$, is there an interpretation of $p^x$?

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    As you are probably well aware, this is a surprisingly deep problem and the answer to most versions of the question is **no**. Googling `Bernoulli factory` provides a significant portion of the relevant literature, among which seminal papers by Keane and by Peres and co-authors.2012-08-15
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    Perhaps this is not the best way to solve the problem, but let p=1/2. This means that for any combination of possible results from any number of tests, the probability of those results can be written as x/2^y, where x and y are whole numbers. However, sqrt(1/2) is irrational.2012-08-15
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    did: I'm not well aware that the problem is deep, I just thought about it and did not find useful references. Your `Bernoulli factory` reference is interesting, it would be frustrating if there is indeed no simpler interpretation for the simple case of the square root. PhiNotPi: Yes, this is a good argument against the existence of a simple interpretation...2012-08-15
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    The probability of success in half a trial?2012-08-15
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    Ross Millikan: I thought of this, but I couldn't understand exactly what it should mean.2012-08-15
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    I don't think that the rationality/irrationality argument is enough to dismiss a simple interpretation. After all, a square with area $2$ has side length $\sqrt{2}$, which no one finds terribly disturbing. (At least, not for the past two-and-a-half thousand years ...)2012-08-15
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    Got something from the answer below?2016-09-02
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    @Did: The answer is interesting and I upvoted it, but I am not accepting it because I don't feel like it fully answers my question. My understanding is that this is a simulation procedure to define a random Boolean variable with probability $1/\sqrt{2}$, but somehow I don't find it as natural as the obvious intuitive interpretation for $p^2$...2016-09-02
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    Did you try to read some of the papers my very first comment is sending you to? If you had, you would know for a fact that what you are waiting for simply does not exist.2016-09-02
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    @Did: Contrary to what your initial comment assumes, I am not familiar with the area, and it is not obvious for me to understand the connection. As far as I can tell, [this article](https://dl.acm.org/citation.cfm?id=175007.175019) and [that article](https://arxiv.org/abs/math/0309222) are about the design of simulation procedures; I'm interested in what an event with probability $\sqrt{p}$ would mean (i.e., how could one interpret it intuitively), not in an effective simulation.2016-09-02
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    @Did: However, if you think that these references are relevant (i.e., that an intuitive explanation would always imply the existence of a simulation procedure, or something like that), it would probably be helpful (to me and to other readers) if you edited your question to explain this in more detail than in your initial comment. :)2016-09-02
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    My question? Which question? Anyway, seeing you hypothetize that references I pointed 4 years ago as relevant (and which **are** relevant, as every expert in the field would confirm) may or may not be relevant is very gratifying, to say the least. Oh, and by the way, your insistence on being given an event of probability $\sqrt{p}$ just shows you did not read the constructions in these papers, which, of course, provide one such event (as every simulation process does), using an unbounded but almost surely finite number of throws.2016-09-02
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    @Did: Apologies, I meant your answer, rather than your question. I did not mean to hurt your feelings and I am not implying anything about your intents, but maybe it is the case that your understanding of my question does not match what I intended to ask. I did not read the above papers as I still fail to understand how they are relevant to what I want. If you think I am mistaken, feel free to edit your answer to explain how they relate. I still cannot understand the connection from your previous remarks.2016-09-02
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    @a3nm " I did not read the above papers" Right. That says it all, I guess. "feel free to edit your answer to explain how they relate" Are you kidding? Bye.2016-09-02
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    "Let us continue this discussion in chat." Mwahaha. Are you for real?2016-09-02

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