2. On 16th August the School of Engineering, Computer and Mathematical Sciences will hosts a Careers Fair. Organisations apply to be part of the fair and if selected are given a stall. Suppose that this year, 32 organisation apply to be part of the Careers Fair, but there is only space for 25. Organisations will be selected at random and each has an equal chance of being selected. (a) How many different sets of organisations could be chosen to attend the Careers Fair? (b) Suppose that within the 32 organisations there are 10 which classify themselves as requiring analytics expertise. Let X denote the number of analytics organisations selected. i. What is the probability that within the 25 organisations at the careers fair exactly 8 will require analytics expertise? ii. State the probability mass function p(x) and cumulative distribution function FIX). Hint: Use R to automate the calculations for the various values of x. iii. Compute the expected value and variance of X, i.e., E(X) and VAR(X). (c) Suppose that during the Careers Fair the organisations will be arranged in a single line and that the 25 organisations, which includes 8 organisations requiring analytics expertise and 17 other organisations, will be arranged randomly. (3 marks) (3 marks) (3 marks) (3 marks) i. Ifall the organisations aredistinguishable, inhowmanydifferentwayscould the organisations (3 marks) be arranged? ii. If the organisations are only distinguished by whether they require analytics expertise or (3 marks) not, in how many different ways could the organisations be arranged? iii. What is the probability that the first 8 organisations will all be require analytics expertise? (3 marks) iv. What is the probabilitythattheorganisationsarearrangedin exactly6runs, 3 oforganisations (4 marks) requiring analytics expertise and 3 of other organisations? For example, if A denotes an analytics organisation and O denotes another type of organisation they could be arranged as follows: