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Consider a deck of $52$ cards. Let $X$ be the number of cards that are drawn until the first ace is chosen (e.g., if the first two cards are not an ace and the third card is an ace then $X=2$). Find $E(X)$.

Hint: Let $I_j$ denote be the indicator variable that is $1$ if the $j$th non-ace is chosen before the first ace and $0$ otherwise for $j=1,…,48$.

I don't really know how to approach this problem, the hint makes it sound more confusing to me.


marked as duplicate by joriki probability May 8 '16 at 0:41

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  • $\begingroup$ To approach the problem, do you see a way to express $X$ in terms of the $I_j$? $\endgroup$ – snarfblaat May 8 '16 at 0:29
  • $\begingroup$ One could find the probability that $X=k$ for the various possible $k$ fom $0$ to $48$, and then use that to find an expression for $E(X)$. Simplifying that expression to $\frac{48}{5}$ is quite painful, while indicator random variables give us the answer quickly. There are other approaches to calculating the expectation, but none as quick as the suggested method. $\endgroup$ – André Nicolas May 8 '16 at 0:34