Poker Digest Vol. 4, No. 15, July 13 - 26, 2001
The advent of online poker has led to considerable discussion about the randomness of the deals at various sites. For example, some people have stridently claimed there are far too many occurrences of quads (four-of-a-kind). We are going to take a look at this claim because it is particularly easy to derive the exact probability of the occurrence of a semideal with at least one player making quads.
In hold'em, quads are possible only if the board shows one pair, two pair, three-of-a-kind, a full house or quads. The same holds true for Omaha except, ironically, when the board has quads no player can have quads.
We take the probability for each type of feasible board, multiply by the probability that one or more players has a hand completing the board to quads, and add the resulting probabilities.
The probability of the board itself being quads is 1/4,165 and all the players have quads in hold'em. The probability of the board being a full house is 6/4,165. Then the probability of at least one player having quads in hold'em is 466/1,081, and the probability of at least one player having quads in Omaha is 932/1,081.
The probability of the board being three-of-a-kind is 88/4,165. Then the probability of someone having quads in hold'em is 20/47, while in Omaha the corresponding probability is 40/47.
The probability of the board being two-pair is 198/4,165. The probability of at least one player having quads is 219/11,891 in hold'em and 350/3,243 in Omaha.
The probability of the board being a one-pair hand is 352/833. Then the probability of a player having quads is 10/1,081 in hold'em and 60/1,081 in Omaha. Multiplying the probabilities and summing gives a probability of .014637, or 1-in-68, for a hold'em semi-deal with at least one player having quads.
Similarly, for Omaha the probability is .047809, or 1-in-21.
I suspect some readers will be surprised by these numbers so let's discuss quads in real games, and let's focus on hold'em. Let's consider only quads arising from a player holding a pocket pair. In this case there is a probability of .004798, or about 1 in 208, that some player makes quads.
Now what is the effect of the order in which the board is dealt?
In Part V we saw that there are 20 ways in which the board may be dealt. If a player holds a pocket pair and the semideal leads to this player making quads, then there is a probability of .9 that the player will flop at least one card of the rank of her pair. Thus, in a loose game where most of the players holding any pocket pair see the flop, a semi-deal leading to quads for a player holding a pocket pair will result in such a player making trips or better on the flop 90 percent of the time. She is not folding now! The point is that if a semideal has quads resulting from a pocket pair, in a loose game it is highly likely the quads are realized.
Quads arising from trips on board is a different matter even in a loose game. The reason is that even in a loose game some players will be folding garbage hands like 3-7 offsuit, 2-10 offsuit and so on. Consequently, trips on board will frequently not produce quads. If the trips are of a high rank there is a higher likelihood someone has hit. Sometimes players who play any two suited cards will waltz into quads. The net effect is that I would guess multiplying the semi-deal probability by something like .8 is going to approximate the actual occurrence of quads in a loose game. This still puts us at quads about once every 85 hands in a loose hold'em game.
Let's use 1-in-85 as a working probability for seeing quads in a hold'em game. What kind of fluctuation should surprise us? For example, at Casino Regina about six weeks ago, I saw one player get quads three times in the span of 90 minutes which means three times in about 50 hands. Twice he made them with pocket pairs and once with trips on board.
You must remember when considering low probability events--and a 1-in-85 chance is such an event-- many trials are needed before you can start being confident that the event is not occurring with the expected frequency. For example, over a span of 85 deals, the probability of seeing four or more deals with quads is about .02. Over a span of 850 deals, the probability of seeing 15 or more hands with quads, which is 50 percent more than the 10 we ``expect'', is .083 or about 1-in-12. This indicates some pitfalls when trying to use something like the occurrences of quads as evidence deals are not random. The pitfalls are exacerbated by the subjective nature of many observors' evidence.