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

Total class size is 10; 7 students take Math and 5 students take Science. Only one student takes none of them. What is the probability that student take both Math and Science.

Then I know

$P(M \cap S)= P(M)+F(S)-P(M \cup S)=0.7+0.5-0.9=0.3$, (easily derived from a Venn Diagram)

but just wonder why I can't simply 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 the answers are different?

Thanks.

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?