In reply to Will Sawyer.

Thanks for sharing this explanation, it was a huge help! In a long blog post, I used it as a starting point for an explanation involving several additional steps (e.g., diagonalizing the matrix to work out the equations). If anyone's interested: http://www.untrammeledmind.com/2020/04/dice-probability-median-rolls-to-see-all-6-sides-coupon-collectors-problem/