What is the probability of any item matching with two sets of items with different, random ordering?

Question

A match would be any one of the same item in each set ending up in the same position. For instance, if two decks of shuffled cards are laid out in pairs, one card from each deck, what are the chances that at least one pair will be of identical cards?

I am more interested in how to solve the general problem than a specific answer. Simple combinatorics doesn't seem to do it. It seems to be devilishly recursive.

The actual answer for two decks is about 60% for at least one match. But I am more interested in a general formula for n items in two sets.

Answer 1

Given your example above, you are basically asking what the probability that a card will be drawn. Twice.

So, you have a 1 in 52 chance of pulling a card. Then you have the same 1 in 52 chance of pulling that same card from the other deck. Even though you are laying the cards down as you go, the probability shouldnt change as the cards have been set, so you are looking at the probability that any given card is in the x/52th slot. In both decks.

That equates to a 1.9% chance in each deck. This multiplied by the % of pulling the same card from the other deck (1.9%) gives you roughly .04% probability of placing the same card in the same slot for each of the decks.

Answer 2

dmc is not correct. The probability must change as you lay down cards.

There are 52! permutations of 1 deck

Laying down a pair at a time gives [52!]^2 different sequences.


Out of these, there's only one in which no two cards match.

** that's not true, there's only one in which EVERY card matches giving 52 pairs ***

**** the following analysis is faulty. I'm leaving it because it might be helpful ****

So, let N = [52!]^2

Then P( no match in 52 pairs) = 1/N

So for at least one match it's

P(at least one matching pair) = 1-1/N

Answer 3

certainly not very likely