What if the bank and the player may roll two times each and use the highest score?
(Say that the bank can roll his die n times and use that highest score and that the player can roll her die m times and use the higher score. Is there a general equation that can be used to find the odds for the player winning?)
Did you work out the math for the first question? That's the easiest.
For the second (and third), I'm inclined to think that the odds are the same as the first question.
Then again, I know the math for the first question, and didn't get out my pencil for the second one yet.....
For the second and third, I would think it favors the house more since there is a better chance of a tie.
Look at the chances of tyoing on a coin toss, where heads equal 1 and tails equal 0.
In one roll, your score can be only 1 or 0, and there's a 50% chance of a tie. In ten coin tosses, the possible range of scores is 0 to 10, average of 5. In a million coin tosses, the range is 0 to 1,000,000, average 500,000 and the chances of tying on 500,000 or any number would be miniscule.
For multiple rolls, the house edge is lower, as there is less chance for a tie score the more rolls are used; the final score range is wider. 10 rolls would average 35, and tying on 35 or any number would be harder. The final score range would be 10 to 60.
To me, the question sounds like they will only take their highest individual roll, so the final score range would still be 1 to 6 with 10 rolls. If this is the case, it is very likely that the player and the house will tie with a 6.
Yeah, I get it now.
The more rolls, the greater the chance for a tie.
However, with ties excluded, there always remains an equal chance of either winning.
But when ties are considered a win for the bank, the more rolls, the greater the house edge.
But that brings me back to my original question (and my feeling that this is homework).
Did you do the math for one roll each? That's the easiest to calculate.
The original poster proposed a one-die game, as well as multiple rolls taking only the highest roll.