In the state of Toyotia, the driving test consists of three parts. Candidates have to pass all three parts in order to get their licence. It is observed that pass rates for each part vary depending on whether the candidate is attempting the part for the first time or repeating it after a previous failure. Pass rates are given in the following table. test part number 1 2 3 pass rate on first attempt 3/10 2/10 6/10 pass rate when repeating 4/10 3/10 7/10 Assume that a candidate's performance on any attempt at any test is independent of their performance on any other attempt. Syntax advice: Give your answers to the following questions as exact fractional expressions, not decimals. You do not have to simplify your fractions, for example, (1/2)+(34/56)+(789/10) is an allowable answer. Make sure you include all necessary brackets, for example, (1+2+3)/(4*5*6). (a) Find the probability that a candidate passes all three tests at the first attempt. Do not enter a decimal. Answer: 3/10*2/10*6/10 (b) Find the probability that a candidate passes all three tests in a total of exactly four attempts. Answer: (c) Given that a candidate passes all three tests in four or fewer attempts, find the probability that they repeated test 2. If you wish, you may type pa for the probability in question (a), and pb for the probability in question (b). Answer: For question parts (d), (e) and (f), let be the number of attempts taken by a randomly chosen candidate to pass test 1. All numbers in your answers must still be given as exact fractional expressions. (d) Assuming that k> 2, find the probability that X = k. Your answer should be an exact formula involving the variable k. Do not use any decimals. Answer: P(X = k) = (e) The expected value of X can be written in the form E(X) = a + b(k), k=2 where a is a specific number and b(k) is an expression in k. What are a and b(k)? For any k > 2, you may write pk for the probability that X = k. Answer: the number a is and the expression b(k) in terms of k is (f) Calculate the numerical value of E(X), giving your answer as an exact fraction. If you wish, you may use the formula k=1 kxk-1 1 = for