Using Venn diagram: Determine if categories are independent

In a group of 100 college freshmen, 25 lettered in band, 20 were in the honor society, and 5 lettered in band and were in the honor society. Determine if the events "lettered in band" and "in the honor society" are independent.

P(E) 25/100=.25, P(F) 20/100=.2, P(ElF) 5/25=.2 and P(FlE) 5/20=.25, so my answer is - a. Independent. Is this right, thanks for checking my answer.

a. Independent
b. Dependent

\(\displaystyle \
\begin{array}{l}
P(E \cap F) = \frac{5}{{100}} = 0.05 \\
\mbox{Does }P(E \cap F) = P(E)P(F)? \\
\end{array}\)

yes

I say yes, but I'm going off my text book to set up the problem just like in the book but with different numbers. The text book shows an example of dependent probabilities. This one I had to do came out the same on both side unlike the text book example. Now my head hurts for real....lol

Since the answer is yes, then they are independent!

thanks

now I feel better.