This is another one of the questions I didn't get a chance to ask the TA at my school today. I thought I had a pretty good grasp on Expectations but apparently I could still use some clarification. Hopefully someone can help me.
Suppose that $n$ people take a blood test for a disease, where each person has probability $p$ of having the disease, independent of other persons. To save time and money, blood samples from $k$ people are pooled together. If none of the $k$ persons has the disease then the test will be negative, but otherwise it will be positive. If the pooled test i sportive then each of the $k$ persons is tested separately (so $k+1$ tests are done in that case)
Let $X$ be the number of tests required for a group of $k$ people. Show that
$ E(X)=k+1-k(1-p)^{k} $