I came across this problem in the Mensa puzzle book cited below.
Book Title: The Mensa Puzzle Book
Author: Victor Serebriakoff (sometime Intl Chairman of Mensa)
Publisher: Treasure press (1991)
At the risk of breaching copyright, I will quote the proble verbatim. I have added items in parentheses for clarity.
Two players shuffle a pack of (playing) cardseach as in snap. What are the chances that they will both lay down the same card at the same point in the deal, a SNAP deal in fact. What are the odds in the dealing of the whole (two) pack(s).
The question is not well posed but I interpret that the author is asking us to calculate the probability that at least one SNAP deal will occur when all 52 cards in each deck are dealt successively.
The author quotes a probility that no SNAP deal will occur. His answer is (51/52)**52
If my interpretation of the problem is correct, the quoted answer appears intuitively to be wrong. However I dont't know what the correct answer is, or how to obtain it!!
I have started my analysis as follows:
Probability of no SNAP = 51/52
There are 51 cards in each deck prior to the second deal, but only 50 of the cards in a deck have a match in the other deck. Accordingly,
Probability of a SNAP deal = (50/51) times (1/51)
Probability of no SNAP = 1 - (50/51.51)
There are 50 cards in each deck prior to the third deal, but either 48, 49 or 50 of the cards in one deck may have matches in the other deck.
I'm stuck now. Is there an easier way to tackle this problem?