Freelance
TravellerDhe
This article originally appeared on the pre-magazine Freelance Traveller website in 1998 and was reprinted in the March/April 2020 issue. Some minor editing has been done, and the analysis of the odds and the flaw in reasoning was added for the September/October 2024 issue..
The Dhe table consists of six squares where you can place your bets and a croupier to handle the dice and bets. The croupier rolls two standard, fair, six-sided dice inside an inverted cup so that nobody can see the result. Each player bets on a number from 1 to 6.
If neither of the dice shows your number you neither win nor lose. House rules govern to determine whether the bet must “ride” to the next roll, or whether it may be withdrawn completely or moved to a different number. Most common is to require the bet to “ride”.
If only one of the dice shows your number you lose.
If both dice show your number you win, collecting 9 times your bet.
The gamblers reasoning goes like this:
“If none of the dice show my number I neither win nor lose so that’s OK. If one of the dice shows my number there is a 1 in 6 chance the other one will as well so if I get more than six times my money I’ll gain on the average.”
Most Dhe tables generate quite a lot of cash so obviously the reasoning above is flawed.
Analysis
The reasoning is indeed flawed; the house has an advantage of 11% on the Dhe tables.
Rolling two fair six-sided dice yields 36 possible outcomes, per the table below.
| Possible rolls of two fair six-sided dice | |||||
|---|---|---|---|---|---|
| 1,1 | 1,2 | 1,3 | 1,4 | 1,5 | 1,6 |
| 2,1 | 2,2 | 2,3 | 2,4 | 2,5 | 2,6 |
| 3,1 | 3,2 | 3,3 | 3,4 | 3,5 | 3,6 |
| 4,1 | 4,2 | 4,3 | 4,4 | 4,5 | 4,6 |
| 5,1 | 5,2 | 5,3 | 5,4 | 5,5 | 5,6 |
| 6,1 | 6,2 | 6,3 | 6,4 | 6,5 | 6,6 |
Assume that the player bets on 1. If neither of the dice shows a 1, the bet is a “push”, and house rules govern. These outcomes (25 of 36 possibilities) are shown with white backgrounds on the table.
If the dice both show 1s, the player wins. This is the outcome (one of 36 possibilities) shown with the green background.
The other possibilities (ten of 36 possibilities), where one die shows 1 but the other does not, are losses. These are shown with red backgrounds. Thus, the payoff for the gambler to ‘break even’ over the long run should be 10:1, not the 9:1 actually offered.
The gambler who tries to ‘track’ rolls to compute actual odds (vs.
offered odds) can get fooled into the 6:1 odds; this is because the
two dice are visually identical. However, if the experiment is
performed “at home” with dice of different colors (red and blue), one
will see that half of the losing rolls are showing the bet number on
the red die, and the rest on the blue die.
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