I need some help seeing if I understand independence and conditional probability.
Let's say I'm expecting a package. It's priority mail so I know at most it will take $3$ days to come. The probability of getting it in the first day $\Pr(A)$ is $0.\overline{33}$. The probability of getting it the second day given it didn't arrive the day before $\Pr(B \mid \neg A)$ is $0.50$ (right?). And the probability of getting it in the $3$rd day given not day $1$ or day $2$ $\Pr(C \mid \neg A \wedge \neg B)$ is $1$ (right?).
So if my logic is correct so far, if I investigate independence, I will find: $\Pr(B \mid \neg A)$ is $0.5$ and $\Pr(B)$ is $0.\overline{33}$; therefore, the events are not independent. Same applies for the third day.
In English, not getting a package in a particular day influences the probability of getting it in the other two days by improving the odds, which intuitively makes sense. Or am I off the mark?
Showing that $P(B)\neq P(B\mid\lnot A)$ proves dependence of $B$ and $\lnot A$, and you have done this nicely. However, to me it's not the most intuitive way to go about it.
If you get a package on one day, then you will definitely not get it on any other day. That sounds very dependent.
More rigorously, another way to define independence of $A$ and $B$ is that $A$ and $B$ are independent iff $$ P(A\text{ and }B)=P(A)\cdot P(B) $$ Is this equality fulfilled?