Suppose I have a deck of cards. The probability of picking the Jack of Hearts is $P(J♥︎)=\frac1{52}$.
Suppose I have an ordinary, unweighted d6. The probability of rolling a $4$ is $P(4)=\frac16$.
Suppose that I seek to predict whether a particular atom will decay by a given time $t$. The probability can be calculated by $P(t)=1-e^{-kt}$, where $k$ is a constant for that atomic species.
These are all described as "random," but to me there's a world of a difference between each of them.
Consider the deck of cards. Even a properly shuffled deck of cards is only considered random due to lack of knowledge by the observer. For instance, suppose for that same deck of cards I already peeked at the top card, and found that it's the Three of Clubs. The probability of picking the Jack of Hearts is $P(J♥︎|3♣︎)=0$. It's exactly the same deck of cards; the physical card doesn't change whether you look at it or not. You might call this ignorant randomness, but I fail to see a difference between not knowing the top card of a deck and not knowing how a computer takes a given seed to create the illusion of randomness. To me, ignorant randomness and pseudo-randomness are different terms for the same idea, that it only appears random due to lack of knowledge of predetermined variables.
Consider the die. Strictly speaking, if one knows exactly the angle and force with which it leaves his hand, and the distance his hand is off the table, and how much and in which direction the die will bounce once it hits, etc., one can predict exactly which number will be rolled. Similar to the deck of cards, a die roll is random only because of lack of knowledge of the observer. Unlike the deck of cards, however, the lack of knowledge is not intrinsic to the event, but rather the system in which it is occurring. The $4$ is not already rolled, the way the Three of Clubs is already on the top of the deck. Any number of minute changes to the system can change the outcome. You might call this extrinsic randomness.
Consider the atomic decay. Quantum mechanics being the weirdness that it is means that it is not possible to predict with $100\%$ certainty when an atom will decay; there are no hidden variables in QM. You might call this intrinsic randomness.
The same way one assigns a value $0\le P\le1$ to describe the likelihood of an event occurring, it seems natural to assign a value $0\le R\le1$ to describe how random the uncertainty is. The same way $P=0$ describes an impossible event and $P=1$ describes a certain event, $R=0$ describes something not random at all and $R=1$ describes intrinsic randomness.
There are some concepts here that overlap with ideas from information theory, measure theory, and chaos theory, but I'm not well-versed enough in the relevant maths to know if what I'm trying to describe can be phrased in terms of those equations. Are there mathematical models out there which express this idea, or which can be repurposed to do so?