a. Paul purchased 8 distinct key chains, 10 distinct photo frames and 12 distinct water globes during a trip as souvenirs for his friends. Suppose Paul selects 3 key chains, 3 photo frames and 3 water globes for his friends at random, calculate the number of selections.
b. In a survey conducted by a travel agency regarding preferred optional places to be visited at country X, 65% of respondents prefer visiting the historical museum, 80% of respondents prefer visiting the palace. Among those respondents who prefer visiting the palace, 30% of them do not prefer visiting the historical museum.
i. Calculate the probability that a randomly selected respondent prefers visiting the historical museum or prefers visiting the palace.
ii. A randomly selected respondent does not prefer visiting the historical museum, what is the probability that he/she does not prefer visiting the palace?
iii. A randomly selected respondent prefers visiting the historical museum, what is the probability that he/she prefers visiting the palace?
iv. What is the probability that a randomly selected respondent who only prefers visiting either historical museum or palace?
c. Three clerks of a consultancy company, Peter, Paul and Mary prepare project reports. From records, the mean numbers of typing mistakes per page made by Peter, Paul and Mary are 1, 2 and 1.5, respectively. It is assumed that the numbers of typing mistakes per page are following the Poisson distribution. Their shares of work in preparing reports are 40%, 25% and 35%, respectively. In final checking, a randomly selected page has more than 2 typing mistakes, calculate the probability that this page is typed by Peter.