Assume our bet is always X.
To win the bet, I roll two dice and aim to make the total greater than my opponent's.
The rules apply to both players
Rule 1. Win twice as much if toss a double.
Rule 2. If tossed a double 6, win instantly without having to wait for the opponent's throw and win three times the bet.
For a normal game that does not have the rules mentioned above, the expected outcome of throwing a die is 3.5, and throwing two will give 7. Since it is a fair die, both players will have an equal expectation of the outcome and a chance of winning the game. So, the expected return will be 0.
My questions:
- How would the results differ if two rules were applied? Assume that the sequence of throws is decided at random (50/50).
- What would it become if I always had the first hand?
Edit:
- If it is a tie, you get your bet back.
- If it is a win, you get your bet back and get rewarded.
- The bet is more like an entry fee, the cost of the ticket for playing this game.
- The gain you make is not directly from your opponent's bet.
- The bet is the base for multiplication decided by the outcome of your throw.
- Win with greater sum has 1x of bet rewarded; With a double tossed, 2x; With a double 6, 3x. Only picking the highest.
- Your gain is the bet times the multiplication mentioned above.
- Whenever a player hit a (6,6), the game ends immediately and that player wins, his opponent will not even have the chance to throw the dice.