# not geometric, not independent, kind of geometric probability problem

Select a card from a standard deck without replacement until you get an ace. Let $X$ denote the number of cards drawn prior to drawing the first ace.

(a) Find the probability distribution for $X$

(b) Use a spreadsheet program to display the probability distribution

(c) Calculate the mean and standard deviation for $X$.

(d) Compare these results to a geometric random variable with $p=4/52$.

So I started this question by drawing a tree diagram then building a table:

$\therefore \Pr(X=k)=\cfrac{P(48,k) \cdot P(4,1)}{P(52,k+1)}$ where P = Permutation Function=P(n,k)

Is there a quick way to calculate E[X] and Var[X] or should I use a computer to tabulate the values for X=0,...,48 and apply the general formulas?

Part (D) I get:

$\Pr(X=k)=(4/52) \cdot (48/52)^k$

$E[x]=(1-p)/p=12$

$Var[X]=(1-p)/p^2=156$

The probability calculation is of course correct. The mean and variance can be obtained by a mildly painful manipulation of binomial coefficients, or by Indicator Random Variable methods. Mean not bad, variance a bit painful. For general information on the negative hypergeometric (which is a slight generalization, waiting time until the $k$-th success) please see this Wikipedia article. – André Nicolas Jul 23 '13 at 1:52