Poker Digest Vol. 3, No. 21, October 6 - 19, 2000
This is the last article in the series dealing with the probabilities of someone having a flush or straight in seven-card stud and hold'em. We have several outstanding issues to settle, followed by a return to the original comment which prompted this series to be written.
The first outstanding issue is a determination of the probability of someone having a straight when seven players are each randomly dealt seven cards. We use inclusion-exclusion just as we did in Part III for calculating the probability that someone receives a flush under these same conditions. However, there are some technical difficulties which we shall not discuss here. It turns out that we use some approximation techiniques and the answer turns out to be approximately .23.
In Part IV in the last issue of Poker Digest, we determined the probability of at least one player having a straight, when 10 players are each dealt two cards and five community cards are turned up in the middle, is about .27. However, in order to arrive at that figure, it was much easier not to eliminate flushes. The second issue we have to deal with is the elimination of flushes. We use the following approximation. In the second article in this series, we saw that there is a probability of .164 that at least one player has a flush under the same conditions. That is, about one in six deals results in a flush. Since the set of deals resulting in at least one straight and the set of deals resulting in no straight are both huge, and since the distribution of suits is more or less independent of the distribution of ranks, the proportion of flushes over these two sets is about the same. Thus, we should remove about one-sixth of the deals which produced straights. This gives us a probability of about .22 that someone has a straight and no one has a flush.
So the probabilities we have are as follows: When randomly dealing two cards to each of 10 players and five community cards, there is a probability of .164 that someone has a flush, and .22 that someone has a straight, but no one has a flush; when randomly dealing seven cards to each of seven players, there's a probability of .185 that someone has a flush, and .23 that someone has a straight, but no one has a flush. Let me reemphasize at this time that we have separated straights and flushes, but we have not eliminated stronger hands. In other words, these probabilities are not the probabilities of the particular hands winning.
This series was motivated by a person who told me that hold'em is a game of straights and flushes, whereas, seven-card stud is a game of two pair. Let's now discuss that comment in light of the mathematical information we have worked out in this series. The probabilities are fairly close, .164 compared to .185 and .22 compared to .23. There are no real differences, but in order to derive those numbers, we worked with idealized situations and approximations in some cases. Perhaps when moving to real games more differences emerge.
One dramatic feature of hold'em is the fact that a player sees 5/7 of her hand for only one bet. Thus, for only one bet, a player can make or need only one card for straights and flushes. This becomes a strong inducement for many players to end up going to the river for flush and straight draws. This is especially true in loose games. There certainly are hold'em games where the incidence of straights and flushes will be fairly close to the above probabilities. Add to this one of those streaks where they are occurring more frequently than expected and it is understandable how someone might view the game as a game of straights and flushes. On the other hand, watch a tight hold'em game and see what you think.
In our idealized setting for seven-card stud we used only seven players, but the game is normally played with eight players. The additional player tends to increase the occurrences of straights and flushes, but the betting structure mitigates against it. Even though seven-card stud has only one more round of betting than hold'em, your hand develops more slowly and this works against people drawing for straights and flushes. So even though both probabilities are higher for the seven-card stud idealized setting, I think the structure of the game reduces the numbers more for seven-card stud.
Still, I think we can draw the conclusion that straights and flushes are in the same ballpark for both games, and the person who originally made the comment to me is probably a victim of selective memory.
On the other hand, Omaha is a...well, maybe another article.