Problem 1:

An optical inspection system is to distinguish among part types. The probability of a correct classification of any part is 0.85. Suppose 4 parts are inspected and that the classifications are independent. Let random variable X denote the number of parts that are correctly classified. Determine the probability mass function of X.

Problem 2:

Samples of emissions from three suppliers are classified for conformance to air quality specifications. The results from 100 samples are summarized as follows:

Let A denote the event that a sample is from supplier 1, and let B denote the event that a sample conforms to specifications. Are events A and B independent? (Only yes or no answer has no points)

Problem 3:

Customers who purchase a certain make of car order an engine in any of three sizes. Of all cars sold 40% have the smallest engine, 35% have the medium-sized one, and 25% have the  largest one. Of cars with the smallest engine, 15% fail an emissions test within two years of purchase, while 10% of those with medium size and 18% of those with the largest engine fail.

1) What is the probability that a randomly chosen car will fail an emission test within two years

2) A record for a failed emission test is chosen at random. What is the probability that it is for a car with a medium engine?

Problem 4:

In a chemical plant, 24 holding tanks are used for final product storage. Four tanks are selected at random and without replacement. Suppose six of the tanks contain material in which the viscosity exceeds the customer requirements.

1) What is the probability that exactly one tank in the sample contain high viscosity material?

2) What is the probability that at least one tank in the sample contain high viscosity material?

3) In addition to the six tanks with high viscosity levels, four different tanks contain material with high impurities. What is the probability that exactly one tank in the sample contains high viscosity material and exactly one tank in the sample contains high impurities?