1. (Bayes theorem) An existing reinforced concrete building must be tested for possible obsolescence. Based on professional judgement, the engineer classifies concrete quality as either x1: 35 to 40, x2: 40 to 45, x3: 45 to 50, or x4: 50 to 60 MPa based on 28-day compressive strength of concrete cubes. The relative likelihoods assigned to these four states x1, x2, x3 and x4 are 0.2, 0.3, 0.4 and 0.1, respectively. Concrete cores are to be cut and tested to help ascertain the true state, although the engineer knows that results from the cores are not conclusive. Therefore, conditional probabilities are estimated to account for the uncertainties involved in examining the cores. These probabilities describe the likelihood that the value of core strength indicated predicts a given unknown state. For example, if the true state is 35 to 40MPa, there is a 70% chance that the tested core strength also lies between 35 to 40 MPa, but there is a 20% chance that it will lie between 40 to 45 MPa, and a 10% chance that it will lie between 45 to 50 MPa. That is, P [sample