Subject: probabilities and Bayes theorem
Details: 1. (Independence of events). The chancellor of a state university is applying for a new position. At a certain point in his application process, he is being considered by seven universities. At three of the seven he is a finalist, which means that he is in the final group of three applicants, one of which will be chosen for the position. At two of the seven universities he is a semifinalist, that is, one of six candidates (in each of two universities).
In two universities he is at an early stage of his application and believes there is a pool of about 20 candidates for each of the two possible positions. Assuming that there is no exchange of information, or influence, across universities as to their hiring decisions, and that the chancellor is as likely to be chosen as any other applicant, what is the chancellor's probability of getting at least one job offer? Please show work, and explain your reason for each step you undertake to solve this problem. The chancellor's faith is in your hand :o)
2.(Bayes Theorem). You're a marketing manger for a nationally renowned company. Before marketing a new product, you need to test it on samples of potential customers. Such tests have a known reliability. For this particular product type, it is known that a test will indicate success of the product 75% of the time if the product is indeed successful and 15% of the time when the product is not successful. From past experience with similar product, you know that the new product has a 60% chance of success on the national market. If the test indicates that the product will be successful, what is the probability that it really will be successful?
Hint:
Let A represent the event of the success of your product.[ based only on your prior assessment or experience from past success with similar product on the national market, before conducting the new test( i.e. before any empirical data obtained)]
Let B represent the event that your test will indicate the product will be successful