Quote from EricP:
The odds of being dealt pocket aces are 1/221.
Note that this is calculated as follows:
Odds of getting an ace on first card = 4/52
Odds of then getting ace on second card = 3/51
Overall odds = 4/52 * 3/51 = 1/221
The odds of being dealt pockets aces twice in a row = 1/48841 (i.e. 1/221 * 1/221)
The odds of being dealt pocket aces three times in a row = 1/10,793,861 (i.e. 1/221 * 1/221 * 1/221)
The odds of being getting a royal flush on the flop is 1/649,740
Calculated as follows:
20/52 * 4/51 * 3/50 * 2/49 * 1/48 = 1/649,740
Note that the first card is 20/52, since any A, K, Q, J or 10 will start the royal flush. After the first card, the suit must match on subsequent cards.
So, to put things in perspective, it is much more rare to have pockets aces three in a row, than even flopping a royal flush. Put another way, if you are a poker addict, and play one hand of poker every minute, 24 hours per day, 365 days per year, you could expect to wait <b>20.5 YEARS of continuous poker play</b> before the first time you receive pocket aces on three consecutive hands.
