Answer to Question #138030 in Combinatorics | Number Theory for T.B.

Question

I have 52 cards (ordinary deck from 2 to Aces, each having four ranks ). I start picking a random card from the deck and add it to the sequence. I stop when the currently selected card belongs to the same rank as the previously selected card. No replacement is allowed. I would like to find what are the probabilities: Selecting the 5th card would be the same rank as the 4th. Or Selecting the 19th card would be the same rank as the 18th. Or None of the selected cards was the same rank as the previously selected card.I know there are 52! possible sequences that can be generated, but the tricky part would be numerator because I do not see any clear pattern.Consider the following sequence:
[ Ace, King, 7, 7 ... n=52] We could say the Probability of the 4th card belongs to the same rank as third is (52 * 48 * 47 * 3)/52!
However:
[7, King, 7,7 .. n=52]. The probability of the 4th card belongs to the same rank as third is (52*48*47*2)/52!
So we can not generalize. Or can we?

Answer 1

We need to take into account the fact that the thirteen ranks are not wholly independent of each other, but rather have a relationship among the 52 cards they lie within. Namely, we cannot treat the probability that no cards of a given rank end up adjacent as a separate event from same probability with regards to another rank.