I'm having a hard time following the scala.

Do you mind writing out what you're doing in pseudocode or in words? Intuitively, this result doesn't seem obvious to me.

Imagine where you have a one line scenario - you watch which register people go to. Label people who went to registers A,B,C with A, B and C respectively. Then, repeat the situation, but instead of one line, use 3, where everyone labeled A stands in one line, B in the other, and C in the third.

The transactions play out exactly like before. I realize we're dealing with probabilities and expected values, but it's definitely not obvious to me.