Description of Chuck-a-Luck:

This game is played with a wired birdcage that holds three dice. The birdcage resembles an hourglass. The dealer turns over the hourglass and three dice fall to the bottom cage. Whatever faces show on the three dice after they land are the winners.

If you place your wager on any of the six numbered squares known as the "Numbers Bet" you will win according to how many dice show the number you bet on.

Players bet on which face will appear. Say you bet that the dice will show a 2. If one die does, you are paid off at even money; if two dice do, you are paid off at two to one, and if all three dice show a 2, you are paid off at three to one. The casino edge is 7.87 percent. (Author: Frank Scoblete)

A Game for Amateur Mathematicians:

A naive mathematician reasons as follows:
Each number comes up 1/6 of the time. If I bet on 5, it will come up on the first die 1/6 of the time. But there are 3 dice, so a 5 should come up 1/6 + 1/6 + 1/6 = 1/2 of the time. That would make the payoff even, but if more than one 5 comes up, I get even more money. I can't wait to start playing.

There are 6*6*6 = 216 ways for three dice to land. Supposing you bet on 5. Of the 216,
• 125 ways involve no 5's,
• 75 ways involve exactly one 5,
• 15 ways involve exactly two 5's, and
• 1 way involves exactly three 5's.
If you bet $1, then your expected gain/loss is: ((-1)*125 + 1*75 + 2*15 + 3*1)/216 = -17/216 = -0.078703703 ..., that is, a loss of almost 8 cents. In other words, this means that the house percentage on this game is 7.8703703...%

The expected gain/loss for a $1 for normal roulette is: 1*(18/38) + (-1)*(20/38) = -0.05263 ..., or a house percentage of 5.263...% Thus roulette is a much better game to play. (Roulette in Monte Carlo has a house percentage of 2.7027...%)

At least one online version of Chuck-a-Luck pays out as above except that they pay $10 if you bet $1 and your number comes up on all three dice. This gives an expected gain/loss of -10/216 = -.04629629... for a house percentage of 4.629629...% -- even better than normal roulette.