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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?

My attempt:

$A$) 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 $1/3$.

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The first box is empty then we should place $3$ balls in $2$ boxes: we have $2^3=8$ possibilities then the probability is $\frac8{27}$.

For the second question:

$$P(B1/B2)=P(B1\cap B2)/P(B2)=\frac18$$

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