# Probability and Odds when Gambling!

Here is a 'lunch break' problem from a rather old publication.

Devise one set of rules for a dice game, where any number of players and one representative of the bank (mandatory), with one die each, can be playing, and where the players and the representative can roll their respective dice any number of times. The rules have to be such that the game is attractive to all the players (i.e. the players feel like there is a good chance of them winning), but that the bank would generate a good profit in the long run. In your game, how much would it cost for a player to play the game, and how much would the bank pay out in the case of a win for one player? What are the odds of a player winning, and what is the ‘house edge’?

This seems rather open-ended to me. Can anyone think of some interesting and profitable rules?

• All my rules so far have only been profitable for the players involved. I have been trying to derive a formula that would link the desired profit for the bank, the number of players involved and the number of rolls per player, but so far have only wasted paper! Is such a formula even possible? – Brian Feb 29 '12 at 22:26
• Wasn't this posted yesterday? – David Mitra Feb 29 '12 at 22:33
• What happened to the original question? – PhiNotPi Feb 29 '12 at 22:43
• Brian deleted the previous question after I commented "Yes this is open-ended and you are supposed to use your own imagination" and Brian replied with the same comment he made above. – Henry Mar 1 '12 at 0:47

One thought would be to have a function $f(n,h)$ where $n$ is the number of rolls and $h$ is the highest roll achieved. You would like it increasing in $h$ and decreasing in $n$, thinking of it as a score for the result. Then pay off depending on who is higher, maybe depending on how much, with the house winning ties.