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Unformatted text preview: EC2019 SAMPLING AND INFERENCE, 2011 QUESTIONS IN PREPRATION FOR THE EXAMINATION The examination asks for THREE answers The time allowed is TWO hours 1. Establish what is meant by (a) a pair of mutually exclusive events and (b) a pair of statistically independent events. Prove, with reference to the axioms of probability and the rules of boolean algebra, that P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) . The probability that a smoker who consumes over 40 cigarettes a day will suffer from chronic respiratory illness is 0.4. The probability that he will suffer from heart disease is 0.5. The probability that he will suffer from at least one of these ailments is 0.6 (i) Find the probability that the smoker will suffer from both ailments, (ii) Find the probability that he will suffer from heart disease given that he suffers from chronic respiratory illness. Answer. Let R denote respiratory illness and let H denote heart disease. We have P ( R ) = 4 10 , P ( H ) = 5 10 , P ( R ∪ H ) = 6 10 . (i) The probability that the smoker will suffer from both ailments is P ( R ∩ H ) = P ( R ) + P ( H ) − P ( R ∪ H ) = 4 10 + 5 10 − 6 10 = 3 10 . (ii) The probability that he will suffer from heart disease given that he suffers from chronic respiratory illness is P ( H | R ) = P ( R ∩ H ) P ( R ) = 3 10 . 10 4 = 3 4 . 2. Derive, from first principles, the function expressing the probability of obtaining x successes in n independent trials when the probability of a success in any trial is p . 1 EC2019 SAMPLING AND INFERENCE, 2011 On the third floor of the Metropolitan Hotel there are six guest rooms but only four bathrooms. On average, two guests in five require a morning...
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- Spring '12
- Probability, Probability theory, respiratory illness, chronic respiratory illness