I've been tinkering with making a board game for a few years now and I need some help with calculating odds.

When attacking, the attacker is going to roll a certain number of 3 sided dice. Usually between 1 and 6.

The defender rolls a certain number of 3 sided dice, usually between 1 and 6.

For every die the attacker throws that isn't matched by a defender die, it is counted as one damage.

I need to know the formula for figuring out how many dice don't match. If there is already an online calculator for figuring this out, that would be perfect. If not, I can work with a formula.

Some examples:
If the attacker rolls three 3-sided dice and the defender rolls three 3-sided dice, what are the chances none will match? what are the chances two of the three Attack dice won't match? If I add one die to the attacker, how does that change the probabilities? If I instead add 1 to the defender, how does that change the probabilities? What if we started with 5 dice for the attacker and 2 for the defender? What if we add 1 to one side?

I understand on a general level that each die thrown has a 33% chance of matching a die on the other side, but when you get multiple dice throws (each die has a 33% chance of being the match) it gets to be too much for me to do by hand.

Bonus points if it is a formula I can stick into excel/google sheets.

- Paul

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