R=1

Question 1 (25 marks) (a) Let A and B be two events. Given the following probabilities for these events: P(A) =0.75, P(B|A) =0.6 and P(BnA) =0.1. (i) What is the probability of P(BnA) ? (ii) What is the probability of P(B) ? (iii) What is the probability of P(A UB) ? (iv) What is the probability of P(A|B)? (v) What is the probability of P(A B)? (10 marks) (b) Let x,, x2 and x, be a positive integer. + = 20, Consider a system of linear equations X 2 + + 2x, + 5x, = 80. Using Gauss-Jordan elimination and performing elementary row operations to solve the above system of linear equations and determine ALL the possible combinations of x1, X2 and x3 . (7 marks) + tw2 + W3 1, (c) Consider the system of linear equations where t and m are real twi + 4w2 + 2W3 = m-R, constant. Use elementary row operations to determine a and B in terms of t and m. (2 marks) (ii) Hence, find all the value(s) of t and m, if any, such that the given system of linear equations has 1) unique solution no solution infinitely many solutions. (6 marks) Page 2 of 6