On average, when I pick a random card out of the deck the expected rank is an $8$. This can be seen by listing out the ranks by order and finding the rank in the middle. So, the probability that my card is higher than yours is the probability that you pick a $7$ or less in the deck, keeping in mind that the deck has $51$ cards in after I have picked my card. $$ $$ Since there are $6$ ranks of cards equal to $7$ or less, and there are $4$ suits for each rank of card, then the probability that my card has a higher rank than yours is equal to $$ P = \frac{24}{51} \approx 47\%. $$