A survey of engineering firms reveals that 85% have their own mainframe computer (M), 16% anticipate purchasing a mainframe computer in the near future (B), and 5% have a mainframe computer and anticipate buying another in the near future. Find the probability that a randomly selected firm: (a) has a mainframe computer or anticipates purchasing one in the near future (4 marks) (b) does not have a mainframe computer and does not anticipate purchasing one in the near future (Hint: consider the Venn diagram of the event) (7 marks) 2 (c) has a mainframe computer given that it anticipates purchasing one in the near future (7 marks) (d) anticipates purchasing a mainframe computer given that it does not currently have one (7 marks) 3. (30 marks) "Go" is one of the oldest games in the world. The objective is to control territory by placing pieces called "stones" on vacant points on the board. Players alternate placing their stones. The player using black stones goes first, followed by the player using white stones. The following table shows the result of a study about the advantage of playing first (i.e., using the black stones) in Go. Number of Wins Number of Games Black Player Level Opponent Level T T 64 116 T L 33 34 L T 8 40 L L 15 30 Total: 120 220 In this table, "T" represents the top-level players and "L" represents the low-level players. When a game randomly selected from this study, let W be the event that the black player wins and TT, TL, LT, and LL be the events of different levels of players combinations. (a) Estimate the probability of winning when one plays first if we randomly select a game in the study. (3 marks) (b) Estimate the probability of winning when one plays first for each combination of the levels of players. (4 marks) (c) If we randomly select a game in the study and it is known that the players are at the same level, what is the probability that the black player wins? (3 marks) (d) Suppose there are five players. Two of them are top-level players and three of them are low-level players. Two players are randomly selected to play Go. One of these two players is randomly selected to be the black player. i. What is the probability that the black player wins? (6 marks) ii. Given that the players are at the same level, what is the probability that the black player wins? (Hint: Let A, B and C be events. It is known that An (BUC)=(ANB) U(ANC)) (8 marks) iii. Suppose it is known that black player wins. What is the probability that the black player is ranked lower than the white player? (6 marks) 4. (20 marks) Two years ago, a statistician estimated that the probability of the exis- tence of Superman is 0.18 based on the evidences in the physical world. He wants to improve his estimation by considering the evidences in morality. He estimates that if Superman exists, the probability that Superman prevents evil (that is, evil does not exist) is 0.78. If Superman does not exist, the probability that the evil exists is 0.5. Given the fact that evil exists, what is the revised probability of the existence of Superman? ~ End~