Cards are drawn one after another from a standard 52 card deck until the first spade is drawn

Question

Cards are drawn one after another from a standard 52 card deck until the first spade is drawn. Let the number of necessary draws be represented by X. What is the mean of X?

My professor said that this question looks like a Negative Binomial but is not. I can't think of what kind of distribution it could be. Any hints?

Answer 1

Imagine you have the following setup:

$\square\;{\spadesuit}1\;\square\;{\spadesuit}2\; \square .... \square\;{\spadesuit} 13\; \square $

On an average, each spade will be separated out evenly and we are interested in the pile that's before ${\spadesuit}1$. You have $52 - 13 = 39$ cards left, and $ \dfrac {39}{14}$ cards for each pile. So you would expect to turn $\dfrac{39}{14}$ cards $+ 1$ to get the first spade

Answer 2

Let $E_n$ be the mean when starting with a deck of 13 spade and $n$ non-spade cards. Clearly, $E_0=1$ and $E_{n+1}=1+\frac{n}{13+n}E_n$. Conclude that $E_n=1+\frac n{14}$. You want $E_{39}$.