The Setup

You have a standard 52-card deck with 4 aces. You shuffle the deck thoroughly and then turn over cards one at a time, stopping when you reveal the first ace.

Question:

What is the expected number of cards you will turn over to see that first ace?

Key Insight

The 4 aces are randomly distributed among the 52 positions in the deck. We’re interested in the expected position of the first ace.

Linearity of Expectation

Let \( A_1, A_2, A_3, A_4 \) be the positions of the 4 aces in the shuffled deck.

The expected position of the first ace is the minimum of these 4 random variables.

For \( n \) cards and \( k \) randomly placed distinguished items, the expected position of the first distinguished item is:

\[ E[\text{min}] = \frac{n + 1}{k + 1} \]

For our case:

\[ n = 52,\quad k = 4 \Rightarrow E[\text{first ace}] = \frac{52 + 1}{4 + 1} = \boxed{\frac{53}{5} = 10.6} \]

Final Answer

\[ \boxed{10.6} \]

So, on average, you will turn over 10.6 cards before encountering the first ace.

Reference