Independent events. Throwing a die just once. If a die is thrown once and:
Event A: a one is obtained.
Event B: a two is obtained.
Are these events independent?
I find it confusing since I think independence relates when at least two throws are made.
 A: The intuitive way to think about the independence of events $A$ and $B$ is to ask whether knowing that $A$ happened changes your knowledge about the probability of $B$.
If you rolled a $1$ does that change the probability that you rolled a $2$?
A: Formally, $\mathbb{P}(A \cap B) = \mathbb{P}(\emptyset) = 0$ and $\mathbb{P}(A)\mathbb{P}(B)=1/6*1/6 = 1/36 \neq 0$ so the events $A,B$ are not independent. This is just the mathematics. What about the intuition?
Knowledge about $A$ gives knowledge about $B$ and vice versa (i.e. if you know that $A$ happened, you know that $B$ cannot happen!).
More generally, if we have two events $A,B$ with $A \cap B = \emptyset$ and $0<\mathbb{P}(A), \mathbb{P}(B)< 1$, then $A$ and $B$ cannot be independent.
A: Independence doesn't require more than one throw.
For example:


*

*the outcome is either $2,$ $4,$ or $6.$

*the outcome is etiher $1$ or $2.$
These events are independent.
But the events you've listed cannot be independent since each has probability $1/6$ and their intersection does not have probability $(1/6)^2.$
