Expected value - deck of cards problem Can you tell me if I have solved this correctly?
If we take a random card from a deck and then put the card back in the deck, and then shuffle it, and repeat this $x$ times, what is the expected number of cards that have been drawn at least one time?

So I know the probability of drawing a card is $1$ out of $52$, hence the probability of not drawing a card is $1 - \tfrac{1}{52}$.
If the process is repeated $x$ times then the probability is $\left( 1 - \tfrac{1}{52} \right)^x = \left( \tfrac{51}{52} \right)^x$.
The probability of drawing a card at least one time is $1 - \left(\tfrac{51}{52} \right)^x$.
The expected value is then $52 \left( 1 - \left( \tfrac{51}{52} \right)^x \right)$.

Indeed, that is the way to go.  Indicator Random Variables, and the Linearity of Expectation.
There are 52 cards in the deck.  The probability of drawing a particular card (say card $i$) at least once is $(1-(51/52)^x)$.
So if $X_i$ is the indicator random variable for this event (ie: takes on value 1 when the event occurs, but 0 otherwise), then as this is a Bernoulli random variable, its expectation is the probability for the event: $$\mathsf E(X_i)=(1-(51/52)^x)$$
Which is the identical for all fifty two indicator random variables.
The count of cards that appear at least once is $\sum_{i=1}^{52}X_i$, so by the Linearity if Expectation:$$\mathsf E\left(\sum_{i-1}^{52}X_i\right)=52\,(1-(51/52)^x)$$.
