The real-life odds of being dealt pocket aces are not dependent on how many cards are left in the deck. In contrast, the odds depend only on how many cards are in the deck (52), quantity of aces in the deck (4), and the cards you receive from a particular deck (2 in case of Holdem).
Calculating The Odds
By simple calculation, you will have a 4 in 54 or a 1 in 13 chance of getting an ace initially. After getting that first ace, you will next have a 3 in 51 or a 1 by 17 chance of getting your second ace. This happens because the total number of possible favorable outcomes and total outcomes decreases by 1 each after the first ace gets picked. This means you have a 1 in 13 multiplied by 17 or a 1 in 221 chance to get pocket aces.
Explanation
To begin with, in total there are just 2652 (52*51) distinctive starting hands. This implies that the odds that you are able to get any two particular cards that are in a given order equals precisely 1 in 2652.
We have 12 distinct ways to get dealt AA (AcAs, AsAc, AcAd, AcAd, AsAh, AsAd, AdAc, AdAh, AhAc, AdAs, AhAs, AhAd). Here, the probability of getting 12 AA in 2652 becomes equal to 1 part per 221 parts. This means you have a 1/221 or 0.4525% chance of receiving pocket aces dealt in any possible way in Holdem.
In addition, the probability of getting dealt pocket aces could never be more than 1/221. In other words, 1/221 is the lowest number that answers the probability, not the highest. Whereas the rest of the numbers such as 1/138 or 1/181 are greater than our answer. These come to represent a greater probability.
Wrapping Up
The odds that you are dealt pocket aces are not dependent on the remaining cards. It is, in fact, based on the total number of playing cards, number of aces, and the cards which you pick from a particular deck.
You will possess a 4 by 52 chance of getting your first ace since there are a total of 4 aces in the entire pack of 2 cards. The probability of getting another ace will be 3 in 51. In this way, the odds that you are getting pocket aces will be 4/52 multiplied by 3/51 or (1/13) x (1/17), which is 1/221 or 0.45%.
