Probability that a player is dealt aces once in the next five deals ? Does it matter if they have aces now?

Question

Someone asked me this today and I am stumped . A player has been dealt aces of any suit combination in a 2 card holdem hand. He/She is holding Aces in their hand. What are the chances the player will be dealt aces of any combination in the immediate next five deals. Cards are being randomly shuffled after each deal.

Answer 1

The chance of getting two aces on a given hand is $\frac 4{52} \cdot \frac 3{51}=\frac 1{221}=p$ and the chance of not getting aces on a given hand is $1-p$. The chance you don't get two aces on any of the next five hands is then $(1-p)^5$ and the chance you get them at least once is $1-(1-p)^5$. How can this hand affect any of the next five if the deck is shuffled?

Answer 2

Prior hand does not effect next 5

Chance of AA is 4 / 52 * 3 / 51 = 1/221

Any of the next 5 is same as 1 - none of the next 5 = 1 - (1 - 1/221) ^ 5 = 2.242% = 1 / 44

Exactly 1 in the next 5 I don't know how to do that