a) A passenger is selected at random from those arriving at a small regional airport in Greece. Let G be the event "the passenger arrived from Germany", U be the

event "the passenger arrived from the UK" and "N" be the event "the passenger arrived from the Netherlands". Let P(G) = 0.35, P(U) = 0.4 and P(N)=0.25.The selected passenger is tested for COVID-19; let P be the event that "the passenger tested positive for COVID-19" and suppose that P(Pos) = 1/10 and P(Pos N U) = 1/7. i) Write down in words what the event N U G is, and find its probability. ii) Write down in symbols what the event "the chosen passenger tested positive for COVID-19 or arrived from the UK" is and find its probability. Show all your calculations using fractions. iii) Show whether events Pos, U are independent or not, by using one of the tests for independence used, given in the list of formulas provided. b) Answer the following. Briefly explain the rationale of your solution for each problem. i) In how many ways can a group of 9 people arrange themselves in a queue? ii) 3 mathematicians and 2 historians sit themselves at a round table. What is the probability that the 2 historians will end up seating next to each other? ii) There are 8 types of pasta available at a supermarket. 5 friends purchase at random one type of pasta each. What is the probability that at least 2 of them will purchase the same type of pasta? c) Is it more likely that we will observe at least one 3 in four rolls of an 8-sided fair die, or that we will observe at least one double 3 in thirty two rolls of two 8-sided fair dice?Provide a brief justification for your solution.

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