I just bumped into a simple question. Let's say I want to compute the probability of taking both Math and Science courses (i.e., $P(M \cap S)$) given this information:

Total class size is 10;
7 students take Math and 5 students take Science.
Only one student takes neither of them. What is the probability that a student takes both Math and Science?

Then I know

$$ \begin{align} P(M \cap S) &= P(M)+F(S)-P(M \cup S) \\ &=0.7+0.5-0.9 \\ &=0.3 \end{align} $$

(easily derived from a Venn Diagram)

but wonder why I can't do $P(M\cap S)=P(M)\times P(S)=0.7\times 0.5=0.35$ in this case, even if $M$ and $S$ seem to be independent events but the result is different). What's the intuition behind the product rule, and why are the answers different?