Question

5. Each of two schools A and B has a recruitment limit of 520 students: it ca number of candidates is equal to 1000, each candidate chooses, independently of the others, the school A or the school B (each choice has probability 1/2). nnot recruit more. The total (3 p.) Find the rough probability that all the students will be recruited to the universities they want. (7 p.) For which value of the recruitment limit, the probability that at least recruited to the desired school is 2 0, 02? one candidate will not be Eoab f throo auasto mou noce tba enlt with

Answer 1

Let X be the mumber of people who choosc school A. Then X Bin(1000, 0.5) (*) The probability of all candidates getting into the school of their choice is Р(X < 520, 1000 — X < 520) — Р(480 < X < 520) Since the number of candidates is quite large enough, hence we can this probability by using Normal approximation to Binomial. Then аpproximate V X 500 X 520) P(-1.265 < Z < 1.265) N (0, 1) P(480 2* 50. 0.8 (*) Observe that for recruitment limit 520, the probability that atleast one candidate will not be recruited to the school thcy desire is 520, 1000- X < 520) ability will increase from the above formula. So the recruitment limit should be ess than 520 for the probability to be > 0.2 1- Р(X < 0.2. Now as we decrease the recruitment limit, the prob S