3. (30 marks) “Go” is one of the oldest games in the world. The objective is to control territory by placing pieces call

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3 30 Marks Go Is One Of The Oldest Games In The World The Objective Is To Control Territory By Placing Pieces Call 1 (161.75 KiB) Viewed 36 times

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. Black Player Level Opponent Level Number of Wins T T 64 T L 33 L T 6 L L 14 Total: 117 Number of Games 116 34 40 28 218 In this table, “T” represents the top-level players and "L” represents the low-level players. (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) = (An B)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)