Total probability and Bayes' theorem for a screening test on water samples

A laboratory uses a rapid screening test to check samples of well water for a contaminant. Its records show that

• \(10\%\) of the samples it receives are contaminated
• a contaminated sample gives a positive result with probability \(0.9\)
• a sample that is not contaminated gives a positive result with probability \(0.05\).

1. A sample is chosen at random. Find the probability that it gives a positive result.[3]
2. A sample chosen at random gives a positive result. Find the probability that this sample is contaminated.[3]