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## Doubling Dice Game

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

 Puzzle ID: #44079 Fun: (2.38) Difficulty: (3.25) Category: Probability Submitted By: javaguru Corrected By: javaguru

You are playing a game of dice with two friends, Tom and Harry. Each person has a cup with six dice. Each round the players ante \$1 each and then all slam their dice cup open-end down on the table. Each player lifts the edge of their cup to look at the roll of their six dice, but keeps their roll concealed from the other players.

On their turn each player may either pass or double the antes for all the players. After the third player's turn all the dice are revealed to determine the winner.

The winner is the person with the most dice with the same value. If two or more players have the same number of matching dice, then the higher value of the dice wins. For example, four ones beat three sixes, but three sixes beat three fives. If two or more players have the same number and value for the winning roll, then they split the antes. The starting person rotates so that the person who went first in a round goes last in the next round.

On the first round you roll a pair of sixes. Harry passed on his turn and now it's your turn. You know that Harry would have doubled the ante if he had three of a kind or better, but that he wouldn't double the ante with a pair when having to go first. So Harry could have the same roll as you, but you're not worried about Harry beating you. You also know that Tom will also double the ante on his turn if he has three of a kind or better, but that he won't double it with a pair.

Should you pass or double the antes?

Bonus question: Does your action change if instead of splitting the antes on a tie, the antes instead stay in the pot for the next round?

### Hint

You need to first determine the probability of each outcome (win/lose/tie) and then determine the expected value for each outcome. The expected value of an outcome is determined by multiplying the probability of the outcome by the net gain/loss associated with that outcome. Add up all the expected probabilities for each outcome to get the expected value for each action.
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