We are given a standard 52-card deck, where ace represents 1, jack represents 11, queen represents 12, and king represents 13. If we randomly shuffle this deck then split it into two decks of 26 cards, what is the expectation of the lowest card value in the left deck (obviously whether we choose the left or right deck here doesn't matter).
My original idea was to use indicator variables, i.e. have $$X_i = \begin{cases}1, \text{ if the smallest card in left deck } \geq i, \\0, \text{ otherwise.}\end{cases}$$ Then $$\text{smallest card in left deck} = \sum_{i = 1}^{13} X_i.$$ Now we can use linearity of expectation to calculate $\mathbb{E}[\text{smallest card in left deck}]$. However, I'm not sure if my solution is correct or if there is another easier way to solve this problem.