Question

Probability questions. Answers are here but I don't know how to do the work so please provide all steps. Thank you!!

1. a) An 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 .4, whereas this probability decreases to .2 for a person who is not accident prone. If we assume that 30 percent of the population is accident prone, what is the probability that a new policyholder will have an accident within a year of purchasing a policy? Answer: 0.26 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? Answer: 0.46154 2. A laboratory blood test is 95 percent effective in detecting a certain disease when it is, in fact, present. However, the test also yields a "false positive" result for 1 percent of the healthy persons tested. (That is, if a healthy person is tested, then, with probability 01, the test result will imply that he or she has the disease.) If.5 percent of the population actually has the disease, what is the probability that a person has the disease given that the test result is positive? Answer: 0.323

Answer 1

1a) P(accident) =P(accident prone)*P(accident|accident prone)+P(not accident prone)*P(accident|not accident prone)=0.3*0.4+(1-0.3)*0.2=0.26

b) P(accident prone|accident)=P(accident prone)*P(accident|accident prone)/P(accident)

=0.3*0.4/0.26=0.46154

2)

P(tested positive )=P(have disease)*P(tested positive|have disease)+P(not have disease)*P(tested positive|not have disease)=0.005*0.95+(1-0.005)*(0.01)=0.0147

therefore P(have disease|tested positive)

=P(have disease)*P(tested positive|have disease)/P(tested positive)

=0.005*0.95/0.0147 =0.323