Each player gets 13 cards. Total number of ways to get 13 cards by each player is [tex] C_{52}^{13}\cdot C_{39}^{13}\cdot C_{26}^{13}\cdot C_{13}^{13} [/tex].
In a draw of 13 cards from a deck having 4 aces and 48 non-aces, player 1 must get exactly 1 ace, the number of ways to do this is [tex] C_{48}^{12}\cdot C_{4}^{1} [/tex]. Now to player 2 we deal 13 cards from a deck having 3 aces and 36 non-aces, and player 2 must get exactly 1 ace and ways to do this are [tex] C_{36}^{12}\cdot C_3^1 [/tex]. Then we deal 13 cards to player 3 from a deck having 2 aces and 24 non-aces, and player 3 must get exactly 1 ace with number of ways to do this [tex] C_{24}^{12}\cdot C_2^1 [/tex]. If players 1-3 each have exactly 1 ace we are done: player 4 will also get one ace. Use the product rule to calculate total number of ways: [tex] C_{48}^{12}\cdot C_{4}^{1}\cdot C_{36}^{12}\cdot C_3^1\cdot C_{24}^{12}\cdot C_2^1 [/tex].
The probability is [tex] Pr=\dfrac{C_{48}^{12}\cdot C_{4}^{1}\cdot C_{36}^{12}\cdot C_3^1\cdot C_{24}^{12}\cdot C_2^1}{C_{52}^{13}\cdot C_{39}^{13}\cdot C_{26}^{13}\cdot C_{13}^{13}} [/tex].
