Consider a balls-and-bins experiment with 2n balls but only two bins. As usual, each ball...

Consider a balls-and-bins experiment with 2n balls but only two bins. As usual, each ball independently selects one of the two bins, both bins equally likely. The expected number of balls in each bin is n. In this problem, we explore the question of how big their difference is likely to be. Let X1 and X2 denote the number of balls in the two bins, respectively. (X1 and X2 are random variables.) Prove that for any e> 0 there is a constant ε > 0 such that the probability .