Suppose you roll 3 white 6-sided dice (d6) and 3 black 6-sided dice. Any black dice numbers cancel out any corresponding white dice numbers. (The multiset of white dice numbers minus the multiset of black numbers)
Example:
- For white dice, you roll (1, 2, 2)
- For black dice, you roll (2, 3, 4)
- The result is (1, 2)
Example:
- White dice: (1, 1, 2)
- Black dice: (2, 5, 6)
- Result: (1, 1)
What is the probability that the result will have any duplicates? (Any repeating numbers)
General Question: For n white-dice, and m black dice, what is the probability that the difference of the white dice group subtracted by the black dice group will have any duplicates?
Let $W$ denote the number of rolls with the white die that result in $1$.
Let $B$ denote the number of rolls with the black die that result in $1$.
Then the probability on a duplicate of $1$ equals:$$P(W=3\wedge B\leq1)+P(W=2\wedge B=0)=P(W=3)P(B\leq1)+P(W=2)P(B=0)$$
This can be found with binomial distribution.
The probability on a duplicate is $6$ times this probability because the events (a duplicate of $1$, a duplicate of $2$, et cetera) are mutually exclusive.