\”Go\” probability

Go is one of the oldest games in the world. The objective is to control territory by
placing pieces called stoned 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 /Number of Games
T_______________ L____________ 30____________ 31
T_______________ T____________ 68____________ 129
L_______________ L____________ 15____________ 26
L_______________ T____________ 7_____________ 40
Total: ________________________ 120___________ 226
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 in Go if we randomly
select a game in the study.
(b) Estimate the probability of winning when you play first in Go for each combination
of player and opponent level.
(c) If we randomly select a game in the study and given that the players are of the
same level, what is the probability that the player with the black stones wins?
(d) Suppose there are five players. Two of them are top-level players and three of
them are low-level players. Two players is randomly selected from them 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?
ii. Given that the players are of the same level, what is the probability that
the player with the black stones win?
iii. Suppose it is known that black player wins. What is the probability that
the black player is ranked lower that the white player?