When is a probability a function of a probability? I am asked to calculate $E(ln (\frac{1}{P(X)}))$ 
Where X is a discrete random variable over
$\chi $ $\in \{1,2,3\}$ with probability mass function 
P(X=1)=0.5,P(X=2)=0.4, P(X=3) = 0.1
I think this would be $\sum_{i=1}^3 P(X=i) ln(\frac{1}{P(X=i)})$
But I am having trouble picturing a real world example.

It looks like you are computing the Shannon entropy, or at least the nats version of it, which is not a probability. For one thing, Shannon's entropy is not bounded from above to unity.
A function of a probability is a probability if it satisfies Kolmogorov's probability axioms. I don't know about a real-world example offhand, but you can find some discussion online about "metaprobability". Closest thing that comes to mind is when a Bayesian defines a prior over the possible values of an unknown probability.
