Engineering Data Analysis

Conditional to Bayes

Problem 1

1. Consider a shopper will choose either brand A or brand B. On each purchase, the probability that he will choose the same brand that he chose on his last preceding purchase is 1/3, and the probability that he will switch brands is 2/3. Suppose that on his first purchase the probability that he will choose brand A is 1/4 and the probability that he will choose brand B is 3/4. What is the probability that his second purchase will be brand B?

Problem 2

2. Suppose that Art can decide to go to work by one of three modes of transportation, car, bus, or commuter train. Because of high traffic, if he decides to go by car, there is a 50% chance he will be late. If he goes by bus, which has special reserved lanes but is sometimes overcrowded, the probability of being late is only 20%. The commuter train is almost never late, with a probability of only 5%, but is more expensive than the bus.

Suppose that Art is late one day, and his boss wishes to estimate the probability that he drove to work that day by car. Since he does not know which mode of transportation Art usually uses, he gives a prior probability of 1/3 to each of the three possibilities.

• What is the boss' estimate of the probability that Art drove to work?
• Suppose that a coworker of Art's knows that he almost always takes the commuter train to work, never takes the bus, but sometimes, 10% of the time takes the car. What is the co-workers probability that Art drove to work that day, given that he was late?

Problem 3

3. Two boxes contain colored balls. Box 1 contains 4 blue, 5 green and 3 red balls. Box 2 contains 6 blue, 2 green and 3 red balls. A ball is selected at random from box 1 and transferred to box 2. A ball is then selected at random from box 2. If the ball selected from box 2 is green, what is the probability that the ball transferred from box 1 was: (a) red? (b) blue?

Problem 4

4. Many diagnostic tests for detecting diseases do not test for the disease directly but for a chemical orbiological product of the disease, hence are not perfectly reliable. The sensitivity of a test is theprobabilitythatthetestwillbepositivewhenadministeredtoaperson whohasthedisease.Thehigherthe sensitivity, the greater the detection rate and the lower the false negative rate.Suppose thesensitivityofadiagnosticproceduretotestwhetherapersonhasaparticulardiseaseis85%.Aperson who actuallyhasthediseaseistestedforitusingthisprocedure bytwoindependentlaboratories.

• The probabilitythat bothtestresultswill be positiveis
• Theprobabilitythatat least one ofthetwotest resultswill be positive

Problem 5

5. Companies provide training to employees for many different reasons. Among them are employeeloyalty, employeeretention,andqualityofemployeework.Suppose65%ofallcompaniesgivesometraining to their employees but that this figure varies by company size. Suppose further that 18% ofall companies using training are small companies and that 75% of all companies that do not usetrainingaresmallcompanies. Acompanyisrandomlysampled without regardto size.

• Whatistheprobabilitythatthecompany usestraining?
• Suppose it isdeterminedthatthe selectedcompanyisnot a small company.
• What istherevised probabilitythat thecompanyusestraining?
• What proportion of allcompaniesisnotsmall?