Consider the problem of distributing $n$ balls into $m$ boxes. Let $X_1, X_2, ..., X_{⌈log2n⌉}$ be a set of independent and uniformly distributed random variables with values in the range $\{1, 2, ..., m\}$. We place the $i$-th ball into the $Y_i=[1+(\sum_{j\in B_{(i−1)}} X_j)\mod m]$-th box, where $B(x)$ represents the set of indices of the bits in the binary representation of $x$ that have a value of $1$. For example, $13=(1101)_2$, so $B(13)=\{1,3,4\}$, which means the $14$-th ball should be placed in the $[1+(X_1+X_3+X_4)\mod m]$-th box. In particular, the first ball is always placed in the first box.
(a) Let $Z$ denote the number of pairs i<j such that balls i and j are placed in the same box. Calculate the mathematical expectation of $Z$.
(b) Prove that when $m=n^2$, the probability that each box contains at most one ball is greater than $1/2$.
It's my homework and I don't have any ideas. Could you provide me with some suggestions?
