# 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.

Added:I didn't have enough information worked out to define the odds. I imagined any number of players, each against the bank (like blackjack). Each player has one die, which he rolls. He can keep the roll or try again. The player score is based on the highest roll achieved and the number of rolls, so if you roll a 6 you quit; if you roll a 1 you try again; if you are in the middle you need to decide. Each player gets a score, then the bank rolls. Bets are settled based on who wins, or maybe the score difference. But that's as far as I went.

To calculate the odds, you define a strategy for the bank. One example would be to roll until you get 5 or 6, then quit. Based on that strategy, you can calculate the probability for the bank of each score. In this case P(f(5,1))=P(f(6,1))=1/6. P(f(6,2))=(4/6)*(1/6) and so on. Then each player can define his strategy, with the best approach to maximize the payoff subject to the known bank behavior. Per the problem, the payoff should be negative, but there should be a number of positive payoffs available.

-
Ross, would you mind explaining your idea further (at the moment, I am not sure that I understand it!)? How would you create this function? –  Brian Feb 29 '12 at 23:24
I would just make a table by experiment. You could use the probabilities that the highest roll in n is h as a guide. The idea would be that at any stage you could settle for your current score or hope to improve it, but if you don't improve your score goes down. I would say f(3,6)<f(1,5) for example because it is easier: 91/216 vs 1/3 –  Ross Millikan Mar 1 '12 at 1:28
Ross, please forgive me if I am missing something obvious here, but what will this 'experimental table' give you? Will this help in the derivation of a formula, or are you simply suggesting one possible set of rules? –  Brian Mar 1 '12 at 1:35
@Brian: I am just suggesting one possible set of rules. As you said, the question is open-ended. This set seems to meet the request that one be able to roll the die as many times as desired and has the feature that one has reason to decide whether to stop or not. You might need to define a stopping algorithm for the bank similar to blackjack. –  Ross Millikan Mar 1 '12 at 4:50
Ross, whenever you have some time, could you please show me how to work out the probabilities, odds and expectation(s) for your example (at the moment, I am still not entirely sure I understand it)? –  Brian Mar 1 '12 at 6:15