On a table there are four standard six-sided dice with faces numbered 1, 2, 3, 4, 5, 6 each. Alice starts and chooses a die. Then Bob chooses a die and both throw the dice. The winner is the person with the greater number. It is a draw, if both show the same number.
Now the numbers on the faces of each of the four dice are changed according to the following rules:
1) number 0 is allowed on a face in addition to the numbers from 1-6.
2) a number can appear more than once on different faces of the same die.
For example you can have a die with faces 0,0,0,4,4,6.
After all four dice have been designed, Alice starts again, chooses a die. (same concept as before). Now Bob chooses a die (he knows which die was chosen by Alice) and both throw the dice.
How must the four dice be designed such that Bob wins with a probability twice as big as that of Alice?