Originally posted by talzamir
Start with two standard decks of 52 playing cards. Shuffle each of them. Then turn open the cards one by one from each deck, going through the whole deck. Find the probability that at some point you'll turn the same card open from both decks, getting for example a pair of jack of clubs.
I wonder if this is equivalent to the (imo far easier) problem of t ...[text shortened]... om each deck, comparing them, returning them to their decks and doing the same again, 52 times.
Depends on the quality of your shuffling
(No, that is actually an interesting point. Most people can't shuffle worth a dime, and most people underestimate how badly they shuffle. That's why casinos are meticulate about specifying how their decks are shuffled - often with a machine instead of a human.)
Assuming a perfect shuffle, the odds are equal to that of shuffling only one of the decks and then doing the same test (this is left as an exercise for the reader
). I don't think it's equivalent to taking a card from each, comparing them, and reinserting them; but it is trivially equivalent to taking a random card from each deck, comparing them, and discarding
them until you run out.
And no, I don't know what the numerical answer is. It should be quite high, though.
1/52, plus 51/52*1/51, plus 1-that times 1/50... ad nauseam. Not doing it. Too many numbers. That\'s what we have computers for.