How awful is the awful magician?

Let $X$ be the random variable denoting the number of attempts made. Note that $Pr(X=k)=\frac{1}{52}$ for every $k\in\{1,2,3,\dots,52\}$. Continue the approach by the definition of expected value.

$E[X]=\sum\limits_{k=1}^{52}kPr(X=k) = \frac{1}{52}\sum\limits_{k=1}^{52}k=\frac{1}{52}\cdot\frac{52\cdot53}{2}=\frac{53}{2}$


There are 52 total cards, and the distribution of the card-of-interest is invariant under reversing the shuffled deck. So the expected value of the number of cards seen (including the card-of-interest) satisfies $x=53-x$, giving $x=53/2$.

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Probability

Expected Value

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