If the proportion of students who obtain failing grades on a foreign University's highly competitive annual entrance examination can be considered as a Beta B(2, 3) random variable,
(i) What is the mode of this distribution, and what percentage of students can be expected to fail annually?
(ii) Determine the probability that over 90% of the students will pass this examina- tion in any given year.
(iii) The proportion of students from an elite college preparatory school (located in this same foreign country) who fail this same entrance examination has a Beta B(1, 7) distribution. Determine the percentage of this select group of students who can be expected to fail; also, determine the probability that over 90% of these elite students will pass this examination in any given year.
(iv) Do these results mean that the elite college preparatory school does better in getting its students admitted into this highly selective foreign University?