The apparent odds in Chuck-A-Luck are very counterintuitive. When I was in high school I showed this game to a very smart math teacher. He couldn't accept that the odds actually favored the house. We then played the game for an hour with the house of course winning and he still thought this was a trick since he knew I did magic.

The reason it works is the probability of not throwing a given number (say six) is (5/6)(5/6)(5/6)= 125/216 (house wins) 57.8%

The probability of throwing one six is 3(5/6)(5/6)(1/6)= 75/216

The probability of throwing 2 sixes is 3(1/6)(1/6)(5/6)= 15/216

The probability of throwing 3 sixes is (1/6)(1/6)(1/6)=1/216

If the game is played 216 times the house will win at 1:1 odds $125

The player will win at 1:1 odds $75, at 2:1 odds $30 {15 times 2), and at 3:1 odds $3, for a total player winnings of $108 compared to the $125 the house won. The house makes $17 on every $216 bet for an edge of 7.87%.

It's interesting that you can come with the same percentage by cubing 5/6. That's because the double and triple odds bring the payoff up to 108 which is half of 216 (instead of 91). Without the double and triple odds the edge to the house is 15.7% (125-91)/216, a real suckers bet.