Problem 1. In a given statistics class, half the students were female. It is also known that ⅔ of the students were business and economics majors, while the rest were psychology majors. It is further known that ⅓ of the girls are Business and economics majors. If I pick a student at random, compute the following probabilities:
That the student is female
That the student is a business and economics major
That the student is a psychology major
That the student is female and psychology major
That the student is a male and business major
Given that the student I pick is a business major, what is the probability that the student is male
Given that the student I pick is a psychology major, what is the probability that the student is female
Are being a business major and being female independent events?
Suppose it is not known that ⅓ of the girls are business majors, but instead that major of students is independent of their gender. Then what fraction of girls are business majors?
Problem 2. A new test for cancer is reported to be 99% sensitive (when tested with people with cancer, it gives a positive result in 99% of the cases) , and 99% specific (when tested with people without cancer, it gives a negative result in 99% of the cases). Assume that 1% of the population is affected by cancer. Given that a person tested positive for the test, what is the probability that he/she has cancer?
Problem 3. Assume that a factory has three machines A, B, C, each producing an identical product. A is responsible for 10% of the production, B for 30% and C for 60% of the production. 20% of the items produced by A are defective. 10% of items produced by B, and 5% by C. What is the probability that any given defective item was produced by C?