This question was given in our first year undergraduate Probability I course.
In $d$ dimensions the lattice points $i = (i_1, i_2, \cdots, i_d)$ where $1\leq i_j\leq n$ may be identified with the “squares” (or, better perhaps, “$d$-dimensional unit cuboids”) of a $d$-dimensional chessboard ranging over $n$ cuboids in each dimension. A $d$-dimensional rook located at the lattice point $(i_1,\cdots , i_d)$ can range freely along points in directions parallel to the coordinate axes (varying $i_j$ , for instance, while keeping $i_k$ for $k\neq j$ fixed). Suppose $r$ rooks are placed at random on the $d$-dimensional chessboard. Say that a given rook is “secure” if it does not lie along the axis-parallel lines of sight of any of the other $r−1$ rooks. Show that the number of secure rooks satisfies an asymptotic Poisson law for a critical rate of growth $r = r_n$ with $n$.
I tried to first find the probability that a rook was secure. We can place the first rook anywhere on the board. Let the first rook be placed at $(i_1, i_2, \cdots, i_d)$. So the next rook can have no coordinates in common with the first rook, for it to be secure. Hence, it has $(n-1)$ options out of $n$ positions for each of the $d$ dimensions. Let $A_i$ be the probability that the $i^{th}$ rook is secure.
$P(A_1)=1$
$P(A_2|A_1)=\left(1-\dfrac{1}{n}\right)^d$
$P(A_3|A_1\cap A_2)=\left(1-\dfrac{2}{n}\right)^d$
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$P(A_r|A_1\cap A_2\cap\cdots\cap A_{r-1})=\left(1-\dfrac{r-1}{n}\right)^d$
$P(A_2)=P(A_1)P(A_2|A_1)+P(A_1^C)P(A_2|A_1^C)=\left(1-\dfrac{1}{n}\right)^d$
$P(A_3)=P(A_1\cap A_2)P(A_3|A_1\cap A_2)+P(A_1^C\cup A_2^C)P(A_3|A_1^C\cup A_2^C)=P(A_1)P(A_2|A_1)P(A_3|A_1\cap A_2)+(1-P(A_1)P(A_2|A_1))P(A_3|A_2^C)=\left(1-\dfrac{1}{n}\right)^d\left(1-\dfrac{2}{n}\right)^d+\left(1-\left(1-\dfrac{1}{n}\right)^d\right)\left(1-\dfrac{1}{n}\right)^d$
I'm unable to proceed in this way to find $P(A_r)$ as the expression becomes very bulky. Also, I don't understand what is meant by "asymptotic Poisson Law for a critical rate of growth $r=r_n$ with $n$".
I appreciate any alternate methods to solve this but it'd be great if someone points out where I'm going wrong in my approach and how can I improve it to get the correct answer.
I already posted this on Mathematics Stack Exchange but received no response. So, I'm crossposting it here.
