Question 2. (4 marks). For each of the scenarios below: i. ii. iii. (a) ('0) Define an appropriate random variable for the scenario by describing it in a sentence. State the probability distribution the random variable you dened in (i) would follow. Use properties of that random variable to perform a calculation or calculations that would be needed to answer the question(s). A fair coin is ipped until heads is obtained exactly 5 times. What is the expected number of tails, and what is the probability that at least 3 tails are flipped? A small business is running a promotion wherein customers who make a purchase are entered into a prize draw. Each customer who makes a purchase has a 1/12 chance of winning a promotional item from the business. During the promotional time, 322 customers make a purchase. Each promotional item will cost the business $5.75, and the business owner has set a budget of $300. What is the expected cost to the business, and what is the probability that the promotion goes over the budget? On an average day, a website has 554 visitors. Each visitor to the website has a 0.1% chance of signing 11p for a paid premium account. What is the probability that no paid premium accounts are signed up for on an average day? You have been asked to monitor a machine that manufactures small parts. The machine builds one part at a time, but due to random chance, an average of 3% of parts are faulty and they cannot be used. Another company has asked you to build 120 of these small parts, and all of them must be working (i.e., not faulty). The machine can produce exactly 40 parts per day (faulty and working combined} before it must be paused for maintenance. If the manufacturing process starts on day 1, what is the expected day that all 120 parts will be reacly for the other company, and what is the probability that the parts will be ready on an earlier day than that? You may complete Question 2 using R, or by hand, or any combination of both. If using R, submit the code you used and the nal answer. If computing by hand, shout all working. You may assume that all trials are independent. You may use a calculator to detenrnine nal answers. Give nal answers to at least 3 decimal places