# Interesting Probability Problem

Here is one for all of the probability geeks out there.  A friend of mine posed this to me over the weekend:

Given a standard shuffled deck of 52 cards, you flip through the deck starting at the top until you reach the first Ace (A).  What position does that A need to be in to give an equal probability of running into another A or a 2 after that?  (I haven't specified whether you have already seen a 2, or not, prior to flipping the first A).

For example, if the first A is the 20th card flipped and I keep flipping the cards, am I more likely to get to another A or a 2 first?  The answer for this is A.

So what is the magic position for the first A so that the probability of flipping an A or a 2 after that is equal.

I originally thought 13.  Well, it turns out that I am wrong (not a surprise).  After multiple computer simulations, it seems that the number is closer 12.  (Actually, it seems to be between 11.5 and 12).

My problem is that I do not know how to write a probability equation for this problem, so I am hoping someone out there could provide some insight.