代做HW2 Conditional Probability and Bayes调试Haskell程序

Conditional Probability and Bayes:

1. An actuary is studying the prevalence of three health risk factors, denoted by A, B and C, within a population of women. For each of the three factors, the probability is 0.1 that a woman in the population has only this risk factor (and no others). For any two of the three factors, the probability is 0.12 that she has exactly these two risk factors (but not the other). The probability that a woman has all three risk factors, given that she has A and B, is 1/3. Calculate the probability that a woman has none of the three risk factors, given that she does not have risk factor A.

2. Insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The company’s statistics show that an accident-prone person will have an accident at some time within a fixed 1-year period with probability 0.4, whereas this probability decreases to 0.2 for a person who is not accident prone. If we assume that 30% of the population is accident prone,

(a)what is the probability that a new policyholder will have an accident within a year of purchasing a policy?

(b)Suppose that a new policyholder has an accident within a year of purchasing a policy. What is the probability that he or she is accident prone?

3. A COVID-19 Rapid Antigen Test is 95% effective in detecting COVID-19 when it is, in fact, present. However, the test also yields a “false positive” result for 1% of the healthy persons tested. (That is, if a healthy person is tested, then, with probability 0.01, the test result will imply that he or she has the disease.) If 0.5% of the population actually has the disease, what is the probability that a person has the disease given that the test result is positive?

4. The following information is given about a group of high-risk borrowers. (1) Of all these borrowers, 30% defaulted on at least one student loan. (2) Of the borrowers who defaulted on at least one car loan, 40% defaulted on at least one student loan. (3) Of the borrowers who did not default on any student loans, 28% defaulted on at least one car loan. A statistician randomly selects a borrower from this group and observes that the selected borrower defaulted on at least one student loan. Calculate the probability that the selected borrower defaulted on at least one car loan.