Question

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?

Answer 1

Solution (a) Define success" a student attends class in lecture hall, ..P[success 0.30 þ (say) - us na 15oo Let or no 0.30 Х the number of students to attend dass in a lecture hall a day out of students. Since the students attend classes independently of other students his/her prior behaniows, is the of suuesses na150o Bernoulli trial with probability of success = in each trial XN Binomial (n=1500 þ = 0.30 Binomial Distribution with parameters n = 1500, þ=0.30. and PME : PIX=a] = (*) p1-1972 120, 1, 2, ., ..E[x] = 1500 X 0.30 = 450 0.30) Thus, ie, brobno m. p - 315 ... var[x] = np(1-5) = np(1-P) = 1500 x 0.308 0.70 there are (6) over flow no of students > no. of seats in a given day. suppose a seats in the lecture hall :. Pr[ overflow ] = Pr. [X>a] :. Pr. I no overflow] = Pr. [xca] our problem is to find as such that Pr. [xca] 0.975 whore X~ Bin (1500, 0.3) we note that for a=484 P[X2 484 2 (M) * (1-p3 0.973439 <0.975 "2 ( 2 ) $* (1.8max -0.9766765 70-975 484 But P[ X= 485 70 1 Thus, hana a = 485 re professor needs 485 seats in lecture I found using qbinom (0.925, 1500,0.3) command in R