This week the question is: Can You Win the Collaborative Card Game?

You and a friend each have a standard deck with 52 cards. You thoroughly shuffle your deck, while your friend thoroughly shuffles theirs. Then, you both draw cards one at a time. If the first card you draw is the same as the first card your friend draws, you lose! Otherwise, you draw again. If the next card you draw is the same as the next card your friend draws, you lose! Otherwise … and so on.

If the two of you can make it through your entire decks without ever drawing the same card at the same time, you both win. Otherwise, you both lose.

What is the probability that you and your friend will win this collaborative game?

Highlight to reveal solution:

I wrote a program to simulate the situation, at least with a deck with 2-9 cards. I reduce to a situation, where deck 1 is simply n numbered cards, going from 1 to n. Then I go through all the permutations of deck 2. There seems to be convergence. I guess the answer is 36.79%. I have no idea why.