Joker Face

Discrete Mathematics Level pending

You were walking inside a mysterious castle, where the Joker was guarding the exit door. He would let you pass if you could win his game.

Afterwards, the Joker held out 2 cards: one with a joker face and one with your face on it. (How did he get that?) He then turned both cards face down on the table and swapped them around, and if you could choose the card with your face at random, you would win and be free to go. On the other hand, if you unfortunately chose the joker face, the Joker would just smile and add in another "blank" card with no faces on it to the combination, and the game would move on. (Obviously, he enjoyed playing with you.)

Now if you happened to choose a blank card, no harms would be done, and the Joker would simply reshuffle the three cards and lay them face down again as if nothing had happened. Also, the same rules applied: if you picked your face card, you won, and if you picked the joker face, the Joker would add another blank card to the game like before. In other words, if you kept on picking up the joker face, the number of blank cards would continuously increase one at a time with the same joker and your face cards remaining in the deck. (In short, there was no way that you were really going to lose, for either Joker genuinely really wanted to help you, or he was just interested in keeping you there forever!)

Supposedly if the Joker had an infinite number of blank cards, what would be the probability of your winning this game? (Assume that he didn't play any tricks behind your back.) Round the answer to the nearest 3 decimal places.

Hint: If you add 2 to the answer, you'll see a very, very famous number.


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