 # Problem 1: In 1994 the US Department of Justice introduced the thre...

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Problem 1: In 1994 the US Department of Justice introduced the three-strikes rule. The rule roughly stipulates that if a person has broken the law two times in the past, then they automatically get a mandatory life sentence if they break the law a third time. Assume that a specific person has a 10% probability of breaking the law each year, independently of other years. Furthermore, assume they can only break the law once in any given year. a) Determine the probability this person will receive a life-sentence exactly 5 years from now. b) Determine the probability this person will receive a life-sentence within the next 5 years. c) Repeat part (a), assuming this person has already broken the law once before. Problem 2: Each of n packages is loaded independently onto either a red truck (with probability p) or onto a green truck (with probability 1-p). Let R be the total number of items selected for the red truck and let G be the total number of items selected for the green truck. a) Determine the PMF, expected value, and variance of the random variable R b) Find the probability that the 1st item to be loaded ends up being the only one on its truck. c) Find the probability that at least one truck ends up with a total of exactly one package. d) Evaluate the expected value and the variance of the difference R - G. e) Assume that n > 2. Given that both of the first two packages to be loaded go onto the red truck, find the conditional expectation, variance, and PMF of the random variable R. Problem 3: A graduating MIE student goes to a career fair, and his/her likelihood of receiving an interview invitation at a booth visit depends on how well he/she did in MIE263 (interviews will be immediately scheduled for a later date). Specifically, an A in MIE263 results in a probability p = 0.99 of obtaining an interview invitation, whereas a C in MIE263 results in a probability p = 0.12 of an invitation. a) Find the PMF for the random variable Y that denotes the number of career fair booth visits a student must make before his/her first invitation (including the visit that results in invitation) Express your answer in terms of p. b) On average, how many booth visits must an A student make before getting an interview? How about a C student? c) Assume that an A student and two C students go to the career fair. If each student can go to at most 5 booths, what is the probability that the A student doesn't get an interview and at least one of the C students gets at least one interview? (Assume the probability of each student getting an interview is independent of the other students.) d) If the A student decides to go to booths until he/she receives 3 interview invitations, Write down the PMF of the number of booths that he/she has to visit. How many booths does he/she have to visit on average? How about a C student? e) Assume that the A student and the C student both visit random booths and get together after each booth visit to share their result. They stop visiting booths when on one round of visits they both get an interview. Let R be the total number of booth visits per student. What is the PMF of R? What is the expected value of R? f) In part (e), Let N be the number of visits until they both have at least one interview. What is the PMF of N? Problem 4: Alice and Bob alternate playing at the casino table. (Alice starts and plays at odd times i=1,3, Bob plays at even times i=2,4, ) At each time i, the net gain of whoever is playing is a random variable Gi with the following PMF: 1/3, g = -2, PG(g) = 1/2, g ==1, 1/6, g = 3. Assume that the net gains at different times are independent. We refer to an outcome of -2 as a loss, and an outcome of 1 or 3 as a win. a) They keep gambling until the first time where a loss by Bob immediately follows a loss by Alice. Write down the PMF of the total number of rounds played. (A round consists of two plays, one by Alice and the one by Bob.) b) Write down the PMF for Z, defined as the time at which Bob has his third loss. c) Let N be the number of rounds until each one of them has won at least once. Find E[N].

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