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This is a weird question that just occurred to me. Assume an event occurs with a definite probability $p$. The given data is a function $f:[0,1]\rightarrow [0,1]$ whose input is a probability for the given event and whose output is the probability that that probability is $p$. My question is whether or not you can use this data to find $p$, and if so, how? My first thought is that you can and that this would be accomplished by integrating the function over the interval, but I'm not sure how to justify this.

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    What do you mean by true probability? – 2017-01-23
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    +navinstudent I suppose I should've began the question with "assume a given event will occur with definite probability" then my question is how to find that probability with the given data – 2017-01-23
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    +navinstudent I just edited it, is my question clear now? – 2017-01-23
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    As I was working on my answer you've edited your question. It makes less sense to me than before. If it has a definite probability $p$, what sense does ' the probability that the probability is p' make? – 2017-01-23
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    Do you want to use the event $A$ to narrow down the possible values of (unknown) $p$? – 2017-01-23
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    +spaceisdarkgreen It means that the probability of the event occuring is an unknown and what you're given is a function that returns the probability that some value on [0,1] is the unknown, this seems to become more and more convoluted the more I try yo explain it. – 2017-01-23
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    @spaceisdarkgreen I just remembered you have to use the @ not the + on SE – 2017-01-23
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    @navinstudent here's a notification (see above) – 2017-01-23
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    Forget about the function (temporarily). Say we have a number $x\in[0,1]$. What do you mean by 'the probability that $x$ is the unknown'? – 2017-01-23
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    @spaceisdarkgreen Statistics is my weakest field so maybe I'm asking a non-sensical question, but the idea is to regard x being equal to p as an event say that the event occurs with some probability. – 2017-01-23
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    What I'm hearing is 'I have a number $p\in[0,1]$, call it the unknown. I also have a probability distribution $f(p)$ for the unknown. Can I use this probability distribution to compute the value of the unknown?' You can compute the mean/mode/median of the probability distribution, but the unknown, err, unknown... – 2017-01-23
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    It totally makes sense to have a scenario where a probability is random. Say you have 10 coins with different probabilities $p_i$ of heads and then pick one randomly. Also, an unknown parameter (like the bias of a coin) can be treated as random for the purposes of statistical inference ( Bayesian statistics). In my answer, I wrote down a theoretical example of such a thing where there are multiple possible true probabilities that are chosen randomly – 2017-01-23
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    It just doesn't make sense to ask 'what its value is' when it's random and our only information is probabilities. If we had data that would allow us to zero in (like in the 10 coins example, say we flipped the coin a bunch and saw it came up heads about 60% of the time) we could infer a better range of possible values for p than the distribution we started out with. But we can't calculate it exactly. – 2017-01-23

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You need a detailed sense of what "the probability that that probability is the true probability for the event occurring" means.

To flesh this out, imagine that the "true probability distribution" (which I'll call a model) is chosen randomly. Imagine we have some discrete set $\mathcal{M}$ of possible models, with a probability distribution $P(M=m)$ for $m\in \mathcal{M},$ where $M$ is a random variable representing what the true model is. Then if your event is $A$ you would have conditional probabilities $P(A|M=m)$ for each possible model $m$. $P(A|M=m)$ is the probability the event $A$ occurs given that the true model is $m$. Then the probability that the event occurs $P(A)$ is given by $$P(A) = \sum_{m\in\mathcal{M}} P(A|M=m) P(M=m).$$

To make contact with your original thoughts, say we have $x = P(A|M=m)$ for some model $m$. We'd like to know 'the probability $x$ is the true probability'. This could be complicated since more than one model might produce this same probability for event $A$. However, if $m$ were the only model that produced a probability of $x$, then we would have $f(x) = P(M=m).$

We can deal with the complication as follows: Since I've been discrete here, assume that $x$ is a discrete variable. Then you'd have $$f(x) =\sum_{m\in\mathcal{M_x}}P(M=m)$$ where $\mathcal{M_x} = \{m\in \mathcal{M} : P(A|M=m) = x.\}$ This just says that 'the probability $x$ is the true probability for $A$' is the probability that the true model has $P(A|M)=x.$

Then we can see that $$\sum_{x}f(x) = \sum_{m\in\mathcal{M}}P(M=m) = 1,$$ since we will just wind up adding up all the $P(M=m).$ So your original idea of just integrating $f(x)$ is off.

But look at $$ \sum_x xf(x).$$ That will weight each $P(M=m)$ by $P(A|M=m)$ and will come out to $P(A).$