So a student of mine is developing a casino game involving rolling dice. The house vs the player. Higher score wins, but the player has to actually beat the house in order to win. Very simple. Things get more interesting when there are more than 2 dice involved. There are various ways of calculating the probabilities of winning different sorts of games, but I really want to know is what is the most efficient way of calculating a problem such as this: What is the probability that if you roll 2 dice, the highest score will be a 3 (for example) then with 3 dice, 4 dice, and better still, what happens with N dice. Just to clarify, this doesn't mean that all the dice are 3 or less. At least one of them has to be a 3. I've found a way of doing it, but it seems very inefficient particularly if N is bigger than 2! For my original question, there are 3 situations if you call the 2 dice A and B: a) dice A is a 3, dice B is less than a 3 b) dice B is a 3, dice A is less than a 3 c) both 3's Now if you had 6 dice, then parts a and b would be similar to calculate. But part c is what's bothering me. You've have to consider ALL of the cases in which at least 2 of the dice would be 3's. You'd be there for ever. That's why I'm convinced there must a far more efficient way of calculating this. So if anyone has any good ideas, please let me know! Thanks very much.