I deal a hand of five card stud.
It is a pair of fives, jack high.
I deal a second hand.
What are the odds it beats the first hand?
The other two cards are technically relevant because the second hand could be 5-5-J-x-x and then we get into second kicker and third kicker situations. That is, unless you are suggesting that all of the cards are reshuffled and the first hand all gets put back in the deck, or the second hand is dealt from a new deck.
EDIT: I also need all the suits, changes Flush probabilities.
The cards in the first hand are the 5 of Hearts, 5 of Clubs, Jack of Diamonds, 8 of Spades, and the 4 of diamonds.
The second hand is dealt from the remaining 47 cards.
If the second hand is also dealt 5, 5, J, 8, 4, the first hand wins.