There is a magician (who is totally not me), who shuffles a standard deck of cards (52 cards, four suits). A volunteer from the crowds chooses a card at random, reinserts into the package, and reshuffles.
The awful magician, being awful, starts from the top of the pack.
"Is this your card?"
"No."
The next card, and the question repeats. On and on, until finally, the volunteer's card reveals itself. Much to the excitement of everyone, that they get to go home.
In this scenario, which is totally not based on real life, what is the expected value of attempts by the awful magician before they manage to bedazzle their audience?
Just to make things slightly more mathematical, the volunteer picks a card at random, then reshuffles the pack. So the question can be recast as given a number $n$ between $1$ and $52$, and a permutation $\pi$ of $\{1,\dots,52\}$, what is the expected value for $\pi^{-1}(n)$?
(The questions comes from playing with a deck of cards, and thinking about Michael Stevens' Vsauce video regarding card tricks, where he cites Scott Czepiel about $52!$.)
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.
$E[X]=\sum\limits_{k=1}^{52}kPr(X=k) = \frac{1}{52}\sum\limits_{k=1}^{52}k=\frac{1}{52}\cdot\frac{52\cdot53}{2}=\frac{53}{2}$