Bell's Theorem - Why does $\lambda$ have a probability? I'm reviewing Bell's theorem. In his proof by contradiction, he assumes the world is deterministic and defines a vector $\lambda$ as the set of all hidden variables which play a role in determining the spin of each entangled electron. He then starts to speak about the "probability density function of $\lambda$", called $\rho (\lambda )$. 
But if the world is deterministic, then $\lambda$ is nonprobabilistic; it is a delta function at precisely the values which the hidden variables take. What is the purpose of introducing a density function at all?
 A: Bell does not assume that the universe is deterministic. He assumes that it is described by a local hidden variable theory. These are not the same thing.
Bell explains this himself by giving an example of a successful hidden variable theory. To briefly recap his argument: take an ensemble of spin 1/2 particles, each in the state $(| \uparrow \rangle+|\downarrow \rangle)/\sqrt{2}$. A measurement of whether the spin of any given particle will be up or down is random and will never be described by a single-valued hidden variable.
However, this system can be described by a local hidden variable theory, if you allow the hidden variable to be distributed with some probability. Bell does exactly this. Specifically, he shows that for a hidden variable that is distributed with an even probability distribution along the hemisphere (on a Bloch sphere) where $\vec{\lambda} \cdot \vec{p}>0$, $\vec{p}$ being the vector describing the spin state of the particle, there is a theory in which this hidden variable correctly reproduces the probability distribution of measurement results for any measurement axis one could choose. In his paper, Bell then goes on to prove that there is no similar theory for more than one particle.
Now, one could imagine that this probability distribution over $\lambda$ is due to a fundamental randomness as in regular quantum mechanics. One could also suppose that it is due to ignorance about the initial conditions of some deterministic system, like the probability distributions that one makes for the outcomes of a coin flip. Bell's inequality is general enough to cover both cases, as long as this hidden variable is local (in the sense that Bell uses this term). The lack of assumptions about $\lambda$ is one reason that this result is so important.
