TopsOJ Summer 2025 - Team Round - Problem 9

Ten fair 10-sided dice (faces numbered 1 through 10) are rolled up to three times. On each roll, any die that shows a prime number is removed and will no longer be rolled. After the third roll and the third round of eliminations of dice has completed, each die that has never shown a prime contributes its face value from the third roll to the total sum; if all dice have been eliminated, the sum is 0. The expected value of this final sum can be expressed as \frac{m}{n} for relatively prime positive integers m and n. Compute m+n.