May I inquire as to which task poses a greater challenge?
The act of correctly predicting the outcome of a single coin toss.
Versus
Achieving victory in Liar's Dice when:
- two players are involved and each player has been provided with two dice.
- two players are involved and each player has been provided with five dice.
- three players are involved and each player has been provided with five dice.
The probability of predicting a single coin toss is 50%. And the probability of winning in Liar's Dice also depends on the strategy of the players and their knowledge and experience. Assuming all players are engaged in a perfect play strategies at Liar's Dice and the coin tosses are fair.
So, which one is harder to predict?
Edit:
In addition, for comparison, please add the breakdown of the difficulty to predict between predicting the above-mentioned Liar's Dice event and rolling a single die, which has a probability of 1/6.
There are some reference links to the rules of Liar's Dice:
By symmetry, the chance of winning any N-player game where all players play perfectly, no ties are allowed, and the player order is randomized, is 1/N. No advantage can be given by strategy since all players play perfectly (meaning identically, given their seat), and any advantage/disadvantage given by player order is exactly balanced out by the fact that the player order is randomized. The probabilities of the first/second/third player winning are ultimately irrelevant when the player order is assigned randomly, since the overall win probability is averaged over player positions and always comes out to 1/N.
If we randomize the player order and everybody plays perfectly, you have a 50% chance of winning the 2-player game, and a 33% chance of winning the 3-player game. It doesn't matter what the game is, so long as there are no ties.