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# Passe-Dix

Our ultimate goal for this quiz is to work out if the game below is fair.

Passe-Dix: Three standard six-sided dice are thrown; players can bet on either manque (scoring 10 or less) or passe (scoring more than 10). If the roll matches their bet, they win the same amount; if not, they lose their bet.

If a player bets manque and rolls a 3, 6, and 1, do they win or lose?

The chart represents the number of ways of rolling a particular sum with three standard dice.

$\begin{array} { | c | c | } \hline \text{ Sum } & \text { Ways to Obtain Sum } \\ \hline 3 & 1 \\ 4 & 3 \\ 5 & 6 \\ 6 & 10 \\ 7 & 15 \\ 8 & 21 \\ 9 & 25 \\ 10 & 27 \\ 11 & 27 \\ 12 & 25 \\ 13 & 21 \\ 14 & 15 \\ 15 & 10 \\ 16 & 6 \\ 17 & 3 \\ 18 & 1 \\ \hline \end{array}$

For example, there are three ways of getting a 4, which can be rolled with $$1+1+2,$$ $$1+2+1,$$ or $$2+1+1.$$

There are $$6^3 = 216$$ rolls possible total. What's the probability of getting a 5 or less?

The chart represents the number of ways of rolling a particular sum with three standard dice.

$\begin{array} { | c | c | } \hline \text{ Sum } & \text { Ways to Obtain Sum } \\ \hline 3 & 1 \\ 4 & 3 \\ 5 & 6 \\ 6 & 10 \\ 7 & 15 \\ 8 & 21 \\ 9 & 25 \\ 10 & 27 \\ 11 & 27 \\ 12 & 25 \\ 13 & 21 \\ 14 & 15 \\ 15 & 10 \\ 16 & 6 \\ 17 & 3 \\ 18 & 1 \\ \hline \end{array}$

What's the probability of getting a manque (10 or less)?

$\begin{array} { | c | c | } \hline \text{ Sum } & \text { Ways to Obtain Sum } \\ \hline 3 & 1 \\ 4 & 3 \\ 5 & 6 \\ 6 & 10 \\ 7 & 15 \\ 8 & 21 \\ 9 & 25 \\ 10 & 27 \\ 11 & 27 \\ 12 & 25 \\ 13 & 21 \\ 14 & 15 \\ 15 & 10 \\ 16 & 6 \\ 17 & 3 \\ 18 & 1 \\ \hline \end{array}$

What's the probability of getting a manque (10 or less) compared to a passe (scoring more than 10)?

Is Passe-Dix a fair game?

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