You have $3$ different balls and you put them at random in potentially $3$ different initially empty boxes meaning any box can hold anywhere between $0$ to $3$ balls (inclusive).
Assume once the ball (or balls) are placed in a box, they will not fall out or be removed during the experiment.
$A$) What is the probability that the first box is empty after all $3$ balls have been placed?
$B$) What is the probability that the first box is empty if it is known that the second box is empty after all $3$ balls have been placed?
$A$) There are $3^3=27$ possible locations for balls. There are $6$ possible outcomes for one box to be empty. $(2,1,0)$, $(1,2,0)$, $(2,0,1)$, $(1,0,2)$, $(0,2,1)$and $(0,1,2)$ and $3$ outcomes for $2$ boxes to be empty: $(3,0,0)$, $(0,3,0)$, $(0,0,3)$.
So, is the probability that the first box is empty = $9/27 = 1/3$? I think that probability should be lower than $1/3$.