Problem
Suppose that 40 percent of the roads in a certain country are concrete and 60% are asphalt. For concrete roads, a certain measurement of surface strength X will be normally distributed with a mean of 35 and variance of 5. For asphalt roads, the surface strength X will be normally distributed with mean 20 and variance of 5. Suppose that a road is selected at random front he population and the surface strength X is measured.
a) Determine the conditional probability that this is an asphalt road given that X = x.
b) For what values of c is the conditional probability in the above item greater than 1/2?