What is intuition behind the product rule of probability and independent events? [duplicate]

Question

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?

Answer 1

Turning my comments into an answer:

These Venn diagrams are a good starting point for understanding how the probabilities of two events interrelate.

If two events are indeed independent, then you can compute $P(A \cap B) = P(A) \times P(B)$. In fact, independence is defined in terms of these joint and marginal probabilities.

In the example you've shown, they're not independent: $M$ and $S$ affect each other. If they were independent, $P(M \mid S)$ would equal $P(M \mid \neg S)$ and $P(M)$. It wouldn't matter what value $S$ took when you consider $M$. But looking at the data, $P(M \mid S) = 3/5$, $P(M \mid \neg S) = 4/5$, and $P(M) = 7/10$. Knowing information about $S$ affects your knowledge of $M$, and vice versa.