Intuition issue with independent events.

Question

Take for example 100 students. Each student studies either French, German, French and German, or none.

Let's say 40 students study French and 25 study German.

If 10 study both then they are independent events, yet if say 9 study both then they are not independent events.

This comes from the formula: $$P(F \cap G)=P(F)P(G)$$

Please draw these two situations using Venn diagrams. I can't for the life of me see why one would be independent yet the other isn't.

Is there any way to ''see'' that one situation is independent or dependent from the Venn diagrams without doing the maths ?

Can someone please give me an intuitive explanation for this?

Answer 1

Independence between $A,B$, imho, is most easy to notice when you look in the same time at $A$ and complement $B$. Conditional probabilities are same, iif events are independent. So $$P(AB)=P(A)P(B) \Leftrightarrow P(A|\overline{B}) = P(A|B)$$

Portion of students studying $G$ among students studying $F$, does not change when you look to students who studying $G$ among students not studying $F$. So studying or not studying $F$ does not influence on portion of students studying $G$. So $$\frac{10}{25}=P(G|F) = P(G|\overline{F})=\frac{30}{75} $$

Answer 2

If the were independant we'd say $40\%$ of the students study French. And $25\%$ of the students study German. And so we'd expect $40\%$ of $25\%$ or $0.40\times 0.25=0.1$ or $10\%$ of the students study both.

So if we find out $10$ students study both and that seems to go along with probability. So they are probably independent.

But if we get an unexpected result such as there are no students who study both (Gee, I guess people who like French hates German and vice versa) or say $24$ study both (Gee, every German student but one study French, I guess there is something about French that appeals to people who like German)... well, then there is something going on and they are not independent.

But we didn't find $10$ students studying both. We found $9$ and that's not what was expected. So the results are independent and there is something (very slightly) repelling students from studying both.

My problem with the intuition is that if this appeared in real life I'd say the sample size is small enough that a discrepancy of $1$ off our expectations is not unusual.

But I think that is what the text is getting at:

If they are independent then $P(F\cap G) = P(F)P(G)= .40\times .25 = 0.1$ and as $P(F\cap G) = 0.09\ne 0.1 = P(F)P(G)$ they are not independent.

And the book is being absolute.