Exercise 2: Crude oil exploration 30pts A newly appointed Minister of Energy believes that his country is bound to have diamonds since all neighboring countries have substantial diamond reserves. After sharing his opinion with a team of engineers, they came up with the prior probabilities of finding diamonds in the country. Those probabilities related to three different types of diamonds are as follows: P (D diamond) = 0.2 D diamonds are colorless P (J diamond) = 0.3 J diamonds are slightly yellow P (Z diamond) = 0.4 Z diamonds are noticeably yellow P (No diamond) = 0.1 a. What is the probability of finding diamonds in the country? 3pts With the technical support of an international oil company, the Minister has launched a nationwide oil exploration program. The oil explorers drilled and extracted samples from different regions. After analyzing one of the samples, they computed the conditional probability of finding that particular sample in the country: P (sample | D diamond) = 0.35 Probability of finding that sample in the country, knowing that it contains D diamonds P (sample | J diamond) = 0.25 Probability of finding that sample in the country, knowing that it contains J diamonds P (sample | Z diamond) = 0.35 Probability of finding that sample in the country. knowing that it contains Z diamonds P (sample | No diamond) = 0.05 Probability of finding that sample in the country, knowing that it does not contain diamonds Page 2 b. What is the probability of finding no diamond in that sample? 3pts c. What is the probability of finding each type of diamond in that sample? 9pts d. What are the posterior probabilities of finding diamond in the country? 12pts e. Compare the posterior and the prior probabilities. 3ptsExercise 2: Beauty Pageant 40pts Miss University is a beauty contest organized by the Inter University Council. The winner of this pageant receives a full scholarship as well as some lucrative commercial deals with prominent beauty and fashion brands and companies. Over 8,000 students compete every year to become Miss University. The competition has five different stages: Stage 1 - Department auditions: 8,000 students compete to represent their department. Stage 2 - Faculty auditions: The 2,000 contestants who have passed Stage I compete to represent their faculty. Stage 3 - University auditions: The 50 candidates who have passed Stage 2 compete for the honor to represent their university. Stage 4 - Live Show: 10 contestants compete for 3 places in the Final. Stage 5 - Final auditions: Where 1 student is crowned Miss University. a. Assuming that all the contestants (8,000) have the same chances, what is the probability of a student selected randomly to be crowned Miss University? 3pts b. Assuming that all contestants have the same chances, what is the probability to advance from Stagel to Stage 27 From Stage 2 to Stage 3? From Stage 3 to Stage 4? And from Stage 4 to Stage 57 12pts Page S After weeks of intense competition and selection, 4 RDU students, 2 NEU students, I CIU student, and 3 EMU students have made it to the Live Show (Stage 4). Calculate: c. The probability to have 2 RDU students in the Final (Stage 5). 5pts d. The probability to have 2 RDU students and I NEU student in the Final. 5pts f. The probability to have at least 1 RDU students in the Final. 5pts g. The probability to have 0 RDU students in the Final. 5pts h. The probability to have 3 RDU students in the Final. SptsExercise 1: Virus outbreak 30pts A statistical study revals that 10% of world population has been infected by a new virus. An electronic portable device is manufactured to identify those who have already been infected but the test is not 100% reliable. It is noticed that if someone is really infected, the test will identify him/her positive in 95% of the cases and if someone is not infected, the test will identify him/her positive in 4% of the cases. A person is selected randomly in the population and tested using the medical device. Let / be the event "the person is infected" 'and T the event "the person is tested positive" a. Evaluate P(/ n T): the probability that a person is infected and tested positive. 3pts b. Evaluate P(/ n T): the probability that a person is not infected and tested positive. 3pts c. Evaluate P(T): the probability of the event T. 3pts d. Evaluate P(/|7): the probability that a person is infected, knowing that the person has been tested positive. 3pts e. Are the events / and 7 independents? 3pts f. A new study reveals that if someone is really infected, the test will identify him/her positive in 90% and not 95% of the cases and if someone is not infected, the test will identify him/her positive in 4% of the cases. Thus, you are asked to repeat the previous questions (a, b, c and d) and revise your probabilities. 15pts