At a certain university, a large course has 1500 registered students; the course can be viewed online if a student does not feel like walking to the lecture hall. Each student decides each day (independently of the other students and independent of her/his own prior behavior) whether to attend the class in the lecture hall or just view the online version. The probability a student attends class in the lecture hall is only 0.30, so the registrar has quite a headache. (a) Let X be the number of students to attend class in the lecture hall on a day. Identify the exact probability distribution of X (its name and parameter(s)). State the expected value and the variance of X. (b) To save money, the registrar wants to use an even smaller lecture hall next semester, but the professor insists on having enough seats so the probability is 97.5% (or more) of no overflow (overflow: the number of students is more than the number of seats) on a given day. How many seats does the professor need in the lecture hall?
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