Cfrgtkky

1 $begingroup$ I am trying to minimise $N$ in the following equation with respect to $b$ . $$N = p^b + (1-p^b)(b+1)$$ Notes: $ 0 le p le 1 $ , because $p$ is a probability. Background: This is my interpretation of a problem concerning how a hospital can minimise the number of HIV blood tests $N$ it has to run by pooling blood samples into bundles $b$ . The HIV test has a probability $p$ of returning negative (no HIV); by extrapolation, the bundle of $b$ samples has a probability of $p^b$ of returning negative. If the bundle returns a positive, each sample in the bundle will be tested individually, yielding $b+1$ tests, the bundle plus each individual sample in the bundle. $N$ denotes the expected number of tests, and is the probability of a bundle negative multiplied by one test, plus the probabil