what is the probability that the waiting time for a randomly selected patient from that clinic is more than 16 minutes

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Question 1 (10 marks) Ronaldo is a football player. Below is his own description about his previous matches: “In one Champions League match, I can score at most 4 goals. The chance to have one goal in a match is four times of having no goal at all. The chance of scoring a brace (two goals) is half of scoring exactly one goal. I am a goal machine. The chance of scoring a hat-trick (three goals) is the same as scoring zero goal in a match. However, it is very rare to have four goals, which is just one-third of the likelihood of scoring a hattrick.” (a) Present the probability distribution of X in a table, where X is the number of goals Ronaldo will score in one Champions League match. (b) Calculate the average number of goals he will score in one Champions League match. (c) Ronaldo takes part in a regular fundraising program. He is donating €2000 for each Champions League match. Moreover, if he scores, he will donate €M for each goal. If he sets a target to donate an average of €3000 in one match. Find the value of M (round off to the integer). Question 2 (20 marks) The waiting time for receiving an occupational therapy treatment in a rehabilitation clinic after arrival is normally distributed with mean 15 minutes and standard deviation 3.5 minutes. (a) What is the probability that the waiting time for a randomly selected patient from that clinic is more than 16 minutes? (b) There are 7% of the patients need to wait for more than k minutes, find the value of k. (c) Suppose there are 8 patients, what is the probability that less than 3 of them need to wait for more than 16 minutes to receive the treatment? Assume their waiting times are independent to each other. (d) Suppose the occupational therapy treatment time in the clinic is normally distributed with mean 30 minutes and standard deviation 5 minutes, which is independent to the waiting time. Find the following summary statistics of the total time a patient spends in the clinic: (di) Mean (dii) standard deviation (diii) Mode

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