Suppose we have two sample spaces

$$ \Omega_{\phi} = \{\phi, \neg \phi \} $$

$$ \Omega_{x} = \{x, \neg x \} $$

with probability distributions $P_{\phi} : \mathcal{P}(\Omega_{\phi}) \rightarrow [0,1]$ and $P_{x} : \mathcal{P}(\Omega_{x}) \rightarrow [0,1]$ respectively. In the table below, I filled in arbitrary values for $P_x$ and $P_\phi$.

\begin{array} {|r|r|r|r|} \hline & \phi & \neg \phi & \\ \hline x &? &? &0.5 \\ \hline \neg x &? &? &0.5 \\ \hline &0.75 &0.25 &1 \\ \hline \end{array}

From the table, we can see that $P_\phi(\phi) = 0.75$ and $P_\phi(\neg \phi) = 0.25$. Likewise, we can see that $P_x(x) = 0.5$ and $P_x(\neg x) = 0.5$.

Question 1: Now I'm trying to understand the meaning of the portions of the table tagged "?". I'm assuming these squares represent the value of the new probability distribution defined on $\Omega_\phi \times \Omega_x$. This would mean that the value of the top left question mark is equal to $P(\phi, x)$. Is this the correct notation and viewpoint?

Question 2: Are the value of the "?" squares now already pre-determined in the sense that the value of the marginal probabilities (arbitrarily defined above) force them to take on certain values already? Put differently, is there more than one solution for the table above?


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