There are n cells in the body, of which cells 1, . . . , k are target cells. Associated with each cell is a weight, with w i being the weight associated with cell i, i = 1, . . . , n. The cells are...

There are n cells in the body, of which cells 1, . . . , k are target cells. Associated with each cell is a weight, with w_i being the weight associated with cell i, i = 1, . . . , n. The cells are destroyed one at a time in a random order which is such that if S is the current set of surviving cells then, independent of the order in which the cells not in S have been destroyed, the next cell killed is i, i ∈ S, with probability wi/Σ_j∈S w_j . In other words, the probability that a given surviving cell is the next one to be killed is the weight of that cell divided by the sum of the weights of all still surviving cells. Let A denote the total number of cells that are still alive at the moment when all the cells 1, 2, . . . , k have been killed, and find E[A].