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PROBABILITY AND STATISTICS, QUIZ 1

(1) Hourly samples of a chemical process are collected for 24 samples per day,but due to the cost considerations only 5 samples are actually analysed eachday. Asume the decision on which 5 samlpes are to be analysed is made atrandom.a) In how many ways can the 5 analysed samples be distributed through theday if the the 5 analyses done are all different?b) In how many ways can the 5 analysed samples be distributed through theday if the 5 analyses done are identical?c) What is the probability that all 5 analysed samples would be from thesamples collected before noon if the analyses done are all different?d) What is the probability that all 5 analysed samples would be from thesamples collected before noon if the analyses done are identical?

(2) Let A, B and C be three events such that A is disjoint from both B abd C,P (A) = 0.2, P (B) = 0.4, P (C) = 0.6 and P (B ∩ C) = 0.3. Determine thefollowing probabilities:a) P (A ∪B ∪ C),b) P (A′ ∩B ∩ C),c) P (A′ ∩B′ ∩ C ′),d) P ((A ∪B) ∩ C),e) P ((A ∪B′) ∩ C ′).

(3) The following table summarizes 204 endotermic reactions involving sodiumbicarbonate.

Final temperature Low heat absorbtion High heat absorbtion266 K 12 40271 K 44 16274 K 56 36

Let A denote the event that a reaction final temperatue is 271 K or less. LetB be the event that the heat absorbed is above target. Let C be the eventthat the final temperature is 271 K or more.a) Determine the conditional probability P (B|A).b) Are the events A and B independent? Justify your answer.c) Are the events A and C independent? Justify your answer.