Conditional Probability: John and Mary's registration cards

You're right that the solution is inconsistent. Already the first sentence in the solution is inconsistent with the problem statement: If the choices for courses are independent events, as stipulated in the problem statement, then students cannot have a fixed number of $2$ courses each. The only way I can see to make sense of this is to assume that the problem statement means only that the choices for/against math and government are independent, whereas the choices for/against other courses can be taken dependent on the math/government choices so as to reach a fixed total of $2$ courses. That would mean that the probability for John’s card to be (other,math) would be $(1-0.3)\cdot0.2$, the probability that he didn't take government times the probability that he took math. As you say, the solution seems to use the value $0.5$ instead of $1-0.3$. I see no valid interpretation that would lead to this value. As you say, it seems that in both cases of cards with another course, the probability for choosing that course has been calculated as the complement of the two probabilities for choosing the other two courses. If so, this is simply wrong.