References for particular definitions of risk and uncertainty
Knight's 1921 essay was not written in formal mathematics (and trying to formulate a direct translation into modern mathematics may be quite problematic). Since Knight's time, a formal decision theory literature has developed which makes distinctions that are at least reminiscent of Knight's.

Lars P. Hansen (2012), writes "Motivated by the insights of Knight (1921), decision theorists use the terms uncertainty and ambiguity as distinguished from risk." Hansen references Gilboa et. al. (2008) who write:


  In economics, Knight (1921) is typically credited with the distinction
  between situations of "risk” and of “uncertainty.” In his formulation,
  “risk” designates situations in which probabilities are known, or
  knowable in the sense that they can be estimated from past data and
  calculated using the laws of probability. By contrast, “uncertainty”
  refers to situations in which probabilities are neither known, nor can
  they be deduced, calculated, or estimated in an objective way.


Gilboa and Schmeidler (1989) introduced a max-min utility theory where instead of simply maximizing expected utility, agents solve a max-min problem where to model ambiguity aversion, the objective is minimized over different priors. This work has been expanded on by Hansen and Sargent in their work on robustness. 

Sliding away from Knight and into the broader topic of decision theory, it would be mandatory to reference Leonard Savage's classic, Foundations of Statistics where he introduces the notion of subjective probability.

Your particular example (in more modern decision theory)

You have a Bernoulli likelihood function for winning the lottery.


To a frequentist statistician, $p$ is a scalar value, a parameter (albeit unknown). There is only one possible outcome for $p$ and so there's no randomness.
If you're a Bayesian in the spirit of Savage, you're willing to extend the tools of probability to model uncertainty in your own mind; you will treat $p$ as a random variable!


If we put a prior on $p$ (eg. the beta distribution is a conjugate prior to the Bernoulli likelihood), we can compute posterior probabilities and make decisions based upon standard expected utility. In terms of how we behave though, there's no difference between a 20% probability that's objective vs. subjective.

Under the ambiguity aversion, max-min model though, we may maximize our control variable over expected utility, taking into account that it will then be minimized over multiple priors: we choose control variable $x$ to maximize utility and then (after observing our choice $x$) a mean guy chooses the prior to minimize utility.

References

Hansen, Lars P., 2012, "Challenges in Identifying and Measuring Systemic Risk", NBER

Hansen, Lars P. and Thomas Sargent, 2001, "Robust Control and Model Uncertainty," American Economic Review

Gilboa, Itzhak and David Schmeidler, 1989, "Maxmin Expected Utility with Non-unique Prior," Journal of Mathematical Economics

Gilboa, Itzhak, Andrew W. Postlewaite, and
David Schmeidler, 2008, "Probability and Uncertainty in
Economic Modeling", Journal of Economic Perspectives

Savage, Leonard Jimmie, 1954, Foundations of Statistics