# Determine Computability of Joint and Conditional Probabilities Given Few Tables

## Problem

I've encountered the following problem during the introduction lecture of my Machine Learning class. I haven't taken a formal probability course yet, so any help would be appreciated.

Consider the multi-valued, random variables: C (campus), G (grade), M (major), and Y (year).

None of these variables are independent. Given the probability tables for the following joint, marginal, and conditional probabilities, explain how to compute each probability below given the probabilities from above, or write “impossible.”

$$P(Y)$$, $$P(M)$$, $$P(G,Y)$$, $$P(C|Y)$$, $$P(C,M)$$, $$P(Y|M)$$

## Attempt

I'm unsure as to how to combine the given probabilities to derive $$P(G | C, M)$$ (#2) and what to specify for $$P(C=main | Y=freshman)$$ given the table for $$P(C|Y)$$ (#3). Does the order of joint and conditional probabilities matter?

1. P(M=compSci, Y=sophomore) = $$P(Y|M)P(Y)$$
2. P(G=B | C=LC, M=business) = ?
3. P(C=main | Y=freshman) = ?
4. P(G=B, M=bio) = not possible

For (2), the only information about $$G$$ that you have is $$P(G,Y)$$, which does not give you any information about the dependence between $$G$$ and $$(C,M)$$. So this is impossible.
For (3), you already have the table for $$P(C \mid Y)$$, so $$P(C=\text{main} \mid Y=\text{freshman})$$ is one entry in this table.