In our Hand Analysis poker forum, one of our members mentions a hand where he had quads against another player's lower quads. He won the hand but the unlikely event triggered his nagging suspicion about online poker being rigged.
Quads are not that extreme in holdem
Quads vs. lower quads, it sure sounds extremely unlikely, but let's see what the odds really are.
In holdem, since you put together your five card hand from a total of seven cards, the chances of having the most unlikely hands like quads or straight flush go up quite a bit compared to the "pure numbers" in many odds tables, where you just draw five cards from the deck.
With seven cards to choose from, the probability of getting quads is about 1 in 595. That's seven times as likely as the 1 in 4200 you're used to see in odds tables. So having quads, while being one of the least likely events in holdem, isn't extremely unlikely, not at all.
Like someone said in the forum, you're bound to get quads once every other poker session or so.
So what about quads versus quads
Okay, but what about another player having a different four-of-a-kind in the same hand? Well, this is another matter, of course. It's not true that this happens once in 600 times given that you have quads. The second four-of-a-kind is much less likely than the first one.
If you have quads and two of the quads cards are in your hand, your opponent has only five cards to use for his quads - two in his hand and the three remaining cards on the board.
With your quad fixed, there are 48 cards left. We must pick five cards from those 48. This results in 1712304 possible five card hands.
Since you've monopolized one of the thirteen available quads, there are twelve quads left to choose from. To any such four-of-a-kind, one more card must be added to form your five card hand, and any of the remaining 44 cards is okay. This gives us 12*44 = 528 possible quad hands, in this situation.
If three of your quad cards are on the board, the opponent has only four cards left for his four-of-a-kind, which gives a total of 12 quad hands. We'll ignore this case for simplicity.
Also not astronomically unlikely
So, the probability for another player to hit quads given that you have quads is 528/1712304 = 0.031%, which corresponds to 1 in 3243.
With several opponents the probability that one of them has quads goes up. On the other hand, the calculations assume that we see the river, which isn't that likely with a paired board. On the other hand, with flopped quads you may very well slowplay your hand quite often, right?
Anyway, as we see, the chances of the highly unlikely event of quads meeting lower quads are in no way astronomically small. Those things are bound to happen in our life-time.
Given the incredible number of hands being played in the online poker rooms, those things will occur. They are in agreement with a fair and well-functioning random number generator.