A medical lab can operate only if at least one licensed x-ray technician is present and at least one phlebotomist. There are three licensed x-ray technicians and two phlebotomists, and each worker is equally likely to show up for work on a given day or to stay home. Assuming that each worker decides independently whether or not to come to work, what is the probability that the lab can operate?

There are a total of \(2^5 = 32\) possible outcomes. From those we should subtract outcomes which don’t contain an x-ray technician or a phlebotomist at all. There are \(2^2\) outcomes that don’t contain an x-ray technician and \(2^3\) outcomes that don’t contain a phlebotomist. Among those two sets of outcomes an outcome which contains no personell at all was counted twice so we need to subtract it to get the total \(2^2 + 2^3 - 1 = 11\) outcomes. The final probability is thus \(\frac{2^5 - (2^2 + 2^3 - 1)}{2^5} = \frac{21}{32}\)