An urn has m black balls. At each stage, a blackball is

Chapter 7, Problem 7.14

(choose chapter or problem)

An urn has m black balls. At each stage, a blackball is removed and a new ball that is black with probability p and white with probability 1 pis put in its place. Find the expected number ofstages needed until there are no more black ballsin the urn.NOTE: The preceding has possible applicationsto understanding the AIDS disease. Part of thebodys immune system consists of a certain classof cells known as T-cells. There are 2 types of Tcells,called CD4 and CD8. Now, while the totalnumber of T-cells in AIDS sufferers is (at least inthe early stages of the disease) the same as thatin healthy individuals, it has recently been discoveredthat the mix of CD4 and CD8 T-cells isdifferent. Roughly 60 percent of the T-cells of ahealthy person are of the CD4 type, whereas thepercentage of the T-cells that are of CD4 typeappears to decrease continually in AIDS sufferers.A recent model proposes that the HIV virus (thevirus that causes AIDS) attacks CD4 cells and thatthe bodys mechanism for replacing killed T-cellsdoes not differentiate between whether the killedT-cell was CD4 or CD8. Instead, it just producesa new T-cell that is CD4 with probability .6 andCD8 with probability .4. However, although thiswould seem to be a very efficient way of replacingkilled T-cells when each one killed is equallylikely to be any of the bodys T-cells (and thus hasprobability .6 of being CD4), it has dangerous consequenceswhen facing a virus that targets only theCD4 T-cells.