1. Define each of the following terms and provide an example for each:

An event, union of events, intersection of events, sample space, mutually exclusive events, the complement of an event, containment, the null event, disjoint events, the probability of an event, probability space, probability by counting, probability of union of events, probability of union of disjoint events, the relation between probabilities of complementary events, relation between the probabilities of the contained event and the probability of its container.

2. Please explain conditional probability accompanied by an example based on a contingency table. Explain how the notion of conditional probability yields the relation for the probability of intersection of independent events.

3. Please explain Bayes’ theorem, accompanied by an example.

4. Demographic characteristics of college students in the USA whose ages are at least 18, according to a survey taken in 1995 are shown in Table 1. Using conditional probabilities, discuss: Â a)The dependence between age and gender, b)The dependence between age and ethnicity for U.S. college students

Table 1

*Demographic characters of college students in the USA whose ages are at least 18 in 1995.*

PART 2:

Please provide your work in detail and include in-text citations.

In the book Making Hard Decisions: An Introduction to Decision Analysis, Robert T. Clemen presents an example in which he discusses the 1982 John Hinckley trial. In describing the case, Clemen says: In 1982 John Hinckley was on trial, accused of having attempted to kill President Reagan. During Hinckleyâ€™s trial, Dr. Daniel R. Weinberger told the court that when individuals diagnosed as schizophrenics were given computerized axial tomography (CAT) scans, the scans showed brain atrophy in 30% of the cases compared with only 2% of the scans done on normal people. Hinckleyâ€™s defense attorney wanted to introduce as evidence Hinckleyâ€™s CAT scan, which showed brain atrophy. The defense argued that the presence of atrophy strengthened the case that Hinckley suffered from mental illness.Â

- Approximately 1.5 percent of the people in the United States suffer from schizophrenia. If we consider the prior probability of schizophrenia to be .015, use the information given to find the probability that a person has schizophrenia given that a personâ€™s CAT scan shows brain atrophy.
- John Hinckleyâ€™s CAT scan showed brain atrophy. Discuss whether your answer in part helps or hurts the case that Hinckley suffered from mental illness.
- It can be argued that .015 is not a reasonable prior probability of schizophrenia. This is because .015 is the probability that a randomly selected U.S. citizen has schizophrenia. However, John Hinckley was not a randomly selected U.S. citizen. Rather, he was accused of attempting to assassinate the President. Therefore, it might be reasonable to assess a higher prior probability of schizophrenia. Suppose you are a juror who believes there is only a 10 percent chance that Hinckley suffers from schizophrenia. Using .10 as the prior probability of schizophrenia, find the probability that a person has schizophrenia given that a personâ€™s CAT scan shows brain atrophy.Â
- If you are a juror with a prior probability of .10 that John Hinckley suffers from schizophrenia and given your answer to part 3, does the fact that Hinckleyâ€™s CAT scan showed brain atrophy help the case that Hinckley suffered from mental illness?Â
- If you are a juror with a prior probability of .25 that Hinckley suffers from schizophrenia, find the probability of schizophrenia given that Hinckleyâ€™s CAT scan showed brain atrophy. In this situation, how strong is the case that Hinckley suffered from mental illness?

Bowerman, B., Drougas, A. M., Duckworth, A. G., Hummel, R. M. Moniger, K. B., & Schur, P. J. (2019). *Business statistics and analytics in practice* (9th ed.). McGraw-Hill

ISBN 9781260187496