Carries, Shuffling, and an Amazing Matrix

For this example, 19/40=47.5% of the columns have a carry of 1. Holte shows that if the binary digits are chosen at random, uniformly, in the limit 50% of all the carries are zero. This holds no matter what the base. More generally, if one adds n integers (base b) that are produced by choosing their digits uniformly at random in {0, 1, . . . , b− 1}, the sequence of carries κ0 = 0, κ1, κ2, . . . is a Markov chain taking values in {0, 1, 2, . . . , n − 1}. The Markov property holds because to compute the amount carried to the next column, one only needs to know the carry and numbers in the current column: the past does not matter. We let P (i, j) = P(κ′ = j|κ = i) denote an entry of the transition matrix between successive carries κ and κ′. Holte found the following: