Basic probability independence in roulette

Question

I am teaching my students how roulette works and we were talking about the probability of betting on the first column of numbers $\{1,4,7,10, 13, 16, 19, 22, 25, 28, 31, 34\}$ and the probability of betting on 2nd $12$, i.e. $\{13-24\}$. In my mind these events should be independent. Thus the probability of landing on the intersection $(13, 16, 19, 22)$ should be $$ P\big(\{13, 16, 19, 22\}\big)=\frac{12}{38}\cdot\frac{12}{38}, $$ but it is clearly $4/38$. Thus I have a contradiction! What am I missing?

Answer 1

Let A be the event of landing on a number in the first column. Let B be the event of landing on the 12-24. Then P(A|B)=1/3, but P(A)=12/38. Thus the events in question are not independent.