Julia Roberts, the chief engineer in charge of testing and evaluations, informed Tom and Mark that the test they were proposing to perform is not in fact 100% accurate. The output of the test is a rating of "Positive", or "Negative". Based on the historical data, Julia shared that whenever the PlanetSaviour system has been installed successfully (i.e. it has performed up to the required specifications), the test has given a rating of "Positive" in 80% of cases and"Negative" in 20% of cases. Similarly, whenever the PlanetSaviour system has been installed unsuccessfully (i.e. it has not performed up to the required specifications), the test has given a rating of "Negative" in 90% of cases and"Positive" in 10% of cases.

1. Calculate the probability that the test produces a rating of "Positive", or "Negative". Also, calculate the probability that an installation of the PlanetSaviour system will be a success (i.e. it will meet the required specifications) given that the test rating is "Positive" or "Negative". Similarly, calculate the probability that an installation of the PlanetSaviour system is unsuccessful (i.e. it will not meet the required specifications) given that the test rating is "Positive" or "Negative". (4 points)
2. How much is the imperfect test worth? Would you recommend it? (4 points)
3. What is a 'Control' lever for risk management? How such a lever is different from the pre-testing lever that you have analysed in parts (c) to (e)? Give an example of Control lever for this setting? (6 points)