Carnival Dice Game : Solution

Out of the 216 equally probable ways the dice may be thrown, you will win on only 91 of them, lose on 125. So your chance of winning at least as much as you bet is 91 / 216, your chance of losing 125 / 216.

If the dice always showed different numbers, the game would be a fair one. Suppose all the squares covered with a dollar. The operator would, on each roll that showed three different numbers, take in three dollars and pay out three. But on doubles he makes a dollar and on triples he makes two dollars. In the long run, for every dollar wagered by a player, regardless of how he places the money, and in what amounts, he can expect to lose about 7.8 cents. This gives the operator a profit of 7.8 percent on each dollar bet.

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