Please Use your keyboard (Don't usehandwriting)

Stat

(1)

An insurance company believes that people can be divided intotwo classes: those who are an accident prone and those who are not.The company’s statistics show that an accident-prone person willhave an accident at some time within a fixed 1-year period withprobability .4, whereas this probability decreases to .2 for aperson who is not accident-prone.
(i) If we assume that 30 percent of the population is accidentprone, what is the probability that a new policyholder will have anaccident within a year of purchasing a policy?
(ii) Suppose that a new policyholder has an accident within a yearof purchasing a policy. What is the probability that he or she isaccident-prone?

(2)

Two boxes containing marbles are placed on a table. The boxesare labeled B1 and B2. Box B1 contains 7 green marbles and 4 whitemarbles. Box B2 contains 3 green marbles and 10 yellow marbles. Theboxes are arranged so that the probability of selecting box B1 is1/3 and the probability of selecting box B2 is 2/3 FATIMAH isblindfolded and asked to select 3 marbles. She will win a color TVif she selects a green marble.
(i) What is the probability that FATIMAH will win the TV (that is,she will select a green marble)?
(ii) If FATIMAH wins the color TV, what is the probability that thegreen marble was selected from the first box?

(3)

One-half percent of the population has CORONA Virus. There is atest to detect CORONA. A positive test result is supposed to meanthat you have CORONA but the test is not perfect. For people withCORONA, the test misses the diagnosis 2% of the times. And for thepeople without CORONA, the test incorrectly tells 3% of them thatthey have CORONA.
(i) What is the probability that a person picked at random willtest positive?
(ii) What is the probability that you have CORONA given that yourtest comes back positive?

(4)

A device is composed of two components, A and B, subject torandom failures. The components are connected in parallel and,consequently, the device is down only if both components are down.The two components are not independent. We estimate that theprobability of:
a failure of component A is equal to 0.2;
a failure of component B is equal to 0.8 if component A isdown;
a failure of component B is equal to 0.4 if component A isactive.
(a)

Calculate the probability of a failure
(i) of component A if component B is down
(ii) of exactly one component
(b)

In order to increase the reliability of the device, a thirdcomponent, C, is added in such a way that components A, B, and Care connected in parallel. The probability that component C breaksdown is equal to 0.2, independently of the state (up or down) ofcomponents A and B. Given that the device is active, what isthe
probability that component C is down?