1-6.5 Two solid-state diodes are connected in series. Each diode has a probability of 0.05 that it will fail as a short circuit and a probability of 0.1 that it will fail as an open circuit. If 44 CHAPTER 1. INTRODUCTION the diodes are independent, what is the probability that the series connection of diodes will function as a diode? 1-6.6 A dodecahedron is a solid with 12 sides and is often used to display the 12 months of the year. When this object is rolled, let the outcome be taken as the month appearing on the upper face. Also let A = (January), B = (Any month with 31 days), and C = (any month with 30 days). Find a) Pr (AUB) b) Pr (AB) c) Pr (CUB) d) Pr(ANC) 1-7.1 In a digital communication system, messages are encoded into the binary symbols 0 and 1. Because of noise in the system, the incorrect symbol is sometimes received. Suppose that the probability of a 0 being transmitted is 0.4 and the probability of a 1 being transmitted is 0.6. Further suppose that the probability of a transmitted 0 being received as a 1 is 0.08 and the probability of a transmitted 1 being received as a 0 is 0.05. Find: a) The probability that a received 0 was transmitted as a 0. b) The probability that a received 1 was transmitted as a 1. c) The probability that any symbol is received in error. 1-7.2 A certain typist sometimes makes mistakes by hitting a key to the right or left of the intended key, each with a probability of 0.02. The letters E, R, and T are adjacent to one another on the standard QWERTY keyboard, and in English they occur with probabilities of Pr (E) = 0.1031, Pr (R) = 0.0484, and Pr (T) = 0.0796. a) What is the probability with which the letter R appears in text typed by this typist? b) What is the probability that a letter R appearing in text typed by this typist will be in error? 1-7.3 A candy machine has 10 buttons of which one never works, two work one-half the time, and the rest work all the time. A coin is inserted and a button is pushed at random. a) What is the probability that no candy is received? b) If no candy is received, what is the probability that the button that never works was the one pushed? c) If candy is received, what is the probability that one of the buttons that work one-half the time was the one pushed?