Assume someone is drawing cards continuously from a deck of cards (without replacement) and stops until he/she gets the 3 or Hearts.
What is the expected minimum rank (Ace = 1, J = 11, Q = 12, K = 13, and so on) among the cards he/she draws?
I think there are three cases to consider: 1) ace before 3 of Hearts 2) ace after 3 of Hearts, but deuce before 3 of Hearts 3) ace and deuce after 3 of Hearts
For case 1) probability is 4/5, 2) is 4/25, 3) 1/25.
Thus, the expected value is 4/5*1 + 4/25*2 + 1/25*3 = 31/25
Is what I am thinking wrong?