Alice, Bob and Carol are hanging around, messing with playing cards.

Alice and Bob each have a new deck of cards, and Alice, Bob, and Carol all know what order the decks are in.

Carol has to go away for a few hours.

Alice starts shuffling the deck of cards with the following weird shuffle: she selects two (different) cards at random, and swaps them. She does this for hours, doing it hundreds and hundreds of times.

Bob does the same with his deck.

Carol comes back and asked “have you mixed up those decks yet?” A deck of cards is “mixed up” if each possible order is approximately equally likely:

\displaystyle \mathbb{P}[\text{ deck in order }\sigma\in S_{52}]\approx \frac{1}{52!}

She asks Alice how many times she shuffled the deck. Alice says she doesn’t know, but it was hundreds, nay thousands of times. Carol says, great, your deck is mixed up!

Bob pipes up and says “I don’t know how many times I shuffled either. But I am fairly sure it was over a thousand”. Carol was just about to say, great job mixing up the deck, when Bob interjects “I do know that I did an even number of shuffles though.“.

Why does this mean that Bob’s deck isn’t mixed up?