PROBLEM 3 (Noah is feeling lucky, 10 points). The Powerball lottery is a game in which you pick 5 distinet \"white ball\" numbers from 1 to 69 and 1 \"red ball\" number from 1 to 26. At the end of the day, the Powerball company randomly picks out 5 distinet white ball numbers and 1 red ball number, all independently of each other and previous days' numbers. The number of correctly guessed balls determines the prize (order does not matter when counting how many you got right). Noah is feeling Iucky, so he decides to buy just one lottery ticket. Answer the following questions: (a) What is the expected number of red ball numbers Noah guesses correctly? (b) What is the expected number of white ball numbers Noah guesses correctly? You may leave vour final solution in summation notation. () Noah decides to buy lottery tickets daily. What is the expected number of days until he guesses a red ball number correctly? PROBLEM 4 {Quang against the Malwares, 10 points). Quang is engineering an Al tool that searches for Malwares installed in computers. This groundbreaking tool operates through a randomized approach, meticulously scanning the operating system in a matter of milliseconds, and succeeds with probability p = 107, If the program does not succeed, Quang runs it again until it succeeds. Each time Quang uses this system its result is independent of the previous trials. In this exercise, we will examine how many trials are needed for the program to succeed. (a) Let Si be the event that the system succeeds at trial i. Clearly, Pr(S;) = 107*. We make n trials, let E, the event that there exists a trial .1