Consider the simplest example, flipping a coin. Is the relationship between the symmetry of the coin and the 50-50 probability synthetic or analytic?
- The probability distribution for a real coin is synthetic (i.e for a given, real coin, it can only be known empirically).
- The probability distribution for a theoretical coin is analytic (or a priori): I.e. it can be derived from the stipulated state probabilities.
- The theory that relates the mathematical distribution to the actual empirical distribution of the coin is what Kant called synthetic a priori. I.e., synthetic truths about the empirical world that come to us a priori. He stipulated this third option to account for Newtonian mechanics.
- If you don’t accept Kant’s third option, then the relationship between the theoretical and empirical should be regarded as empirical—i.e. the connection itself between the mathematical and empirical can only be known empirically. Conceptually, this is just a restatement of the experimental requirement of science. I.e., while the mathematical equations in science might be derived analytically, to know that they are true, we still have to go out into the real world and experimentally test them.
By definition, a fair coin has a 50-50 probability distribution. That "by definition" makes the connection between the term and the probability distribution analytic. The probability distribution isn't derived, however; instead it's stipulated. So we have an analytic probability distribution, but not an analytically derived one yet.
To get an example of an analytically derived probability distribution, consider the random variable the fraction of n independent flips of the fair coin that are heads. Call this random variable H(n). Using our stipulated probability distribution for the fair coin, plus the Kolmogorov axioms for probability, we can derive the probability distribution for H(n) (it's called the binomial distribution).
So now we've used definitions for the fair coin, plus definitions for probability, to derive another probability distribution. So the binomial distribution for H is analytically derived.
This basic approach is widely used in Fisherian statistical hypothesis testing. First you stipulate a null hypothesis, which characterizes how you think your experimental setup will work when things are happening "purely by chance." This corresponds to the fair coin. Then you identify a statistic of interest — this corresponds to H — and derive a probability distribution for that statistic given the setup you stipulated in the null hypothesis. This derived distribution is called the sampling distribution. After conducting your experiments, you compare the observed value of the statistic to the sampling distribution. That gives you a probability p of observing that value, given the null hypothesis. That p is the (in)famous p-value used to determine statistical significance. (Fisher's version of hypothesis testing doesn't involve the idea of statistical significance, though.)
In this approach, the sampling distribution is analytically derived. Then it's compared to the empirical data. Many academic statisticians (in the sense of people with PhDs in statistics) focus their work on understanding the probabilities of sampling distributions and other analytically derived mathematical constructs. They rarely, if ever, work with empirical data. (This is one criticism of academic statistics from a data science perspective.) In that respect, I would disagree with @barrycarter's distinction between probability and statistics.
I feel like the De Re/De Dicto distinction might be helpful here; De Re refers to the object specifically, and De Re refers, in some sense, to the ways the object is said to be (this will become clearer in what follows).
When we say that a fair coin has a 0.5 probability of landing heads (or tails), this can be interpreted in two ways.
On the one hand, there is the de dicto interpretation: it can be understood as a statement about any coin that has the property of being a fair coin. As, in this case, the property of being a fair coin is defined simply as having a 0.5 probability of landing heads (or tails) when flipped. If we take the extension of the property termed being a fair coin, it will return for us the set of all coins that have a 0.5 probability of landing heads (or tails) when flipped, it is an analytic truth of anything that falls within the set of things that have the property of being a fair coin.
Yet, if it is understood as De Re, we find a different result: when understood in this way, we are saying something of the coin (that is, of the entity) that it is fair and thereby has a 0.5 probability of landing heads (or tails). Rather than the set of things that have the property of being a fair coin, it is the actual object that is being predicated over; that is, this property is attributed to the object. For the object to have such a property is synthetic.
So, I suppose, in short, the answer is: it depends how you interpret the sentence.
Firstly, symmetry is only absolute in Math, not in Math-Physics (search for Closer to Truth interview videos on Symmetry). So while putting a physical coin in distribution of flips, it's the jurisdiction of Math-Physics hence no good sense.
Nevertheless, intuitively, the coin symmetry is only one among many parameters such as the surrounding air or liquid, electromagnetic fields, ... that if are stripped off, again makes little sense in terms of truths other than approximation.
Clearly it's analytic.
Yes, the probability of finding an electron at a given distance from its nucleus is analytically derived.
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