In order for a game to be fair, both players must have the same chance of winning. Let's examinate the four games:
First game:
First player wins with 1,2,3, second player wins with 4,5,6. Both have 3 chances of winning, with a probability of 1/6 each. So, both players have a winning chance of 3/6=1/2, and thus the game is fair.
Second game:
First player wins with heads side up, second player wins with tails side up. Both have 1 chances of winning, with a probability of 1/2 each, and thus the game is fair.
Third game:
First player wins with 2,4,6, second player wins with 1,3,5. Both have 3 chances of winning, with a probability of 1/6 each. So, both players have a winning chance of 3/6=1/2, and thus the game is fair.
Fourth game:
First player wins with 3,4,5 or 6, second player wins with 1 or 2. Player 1 has four chances of winning, with a probability of 1/6 each, whereas player 2 has 2 chances of winning, with a probability of 1/6 each. So, player 1 has a winning chance of 4/6=2/3, while player 2 has a winning chance of 2/6=1/3, and thus the game is not fair.
