Poker Probabilities and Mathematics

Down the River or Seven Card Stud

In Australia poker game the average player would like to know what his chances are of making his straights and flushes starting with various combinations.
In studying Table IX the first thing we note is that if we start with three cards of the same suit it is about four and a half to one against our making the flush. Now if our next card is also the same suit, our chance is almost even money; while if it doesn't help us, the chances against us increase to nine to one. Accordingly, if my three cards are all in the same suit, unless the pot is raised I am going to stay and draw a fourth card. But if that fourth card is not in my suit, my participation in that particular pot is ended unless that fourth card happens to give me a pair, in which case I may trail along, depending on just how much strength is shown by the other players. The queen, jack, ten is given as an example of any three cards in succession that may become a straight upon the addition of two cards at either end or one card at each end. Players will do well to note how much less the chance is when starting with king, queen, jack or four, three, two.
It is further worthy of note that, as in the case with three to a flush, if your fourth card does not help you there is little point in staying further since the chances against your eventually making a straight have increased from four to one to eight to one.

Possibilities For High Low

However, in High-Low any three of a kind becomes much less valuable because you have practically no chance whatsoever of winning low and hence the best you can hope for is to take half the pot.
On the other hand, it will be noted that two pairs are by no means anything to get too optimistic about. In the first Place, even if your first four cards are two pairs, the odds are still more than four to one against your eventually making a full house; while with two pairs and an odd card the chances are seven to one against you. Although two pairs may win the pot, remember no guarantee goes with them. Finally, in studying possibilities for High-Low poker, it should be noted that holding a pair and an odd card, your chance of eventually coming out with a full house is less than one in thirteen. Therefore, unless you have possibilities for low, the fact that you have a pair does not justify your staying for a fourth card in a High-Low game.

Remember the Values

Then when looking at an x value, remember that an eight is likely to be of value, and even nines and tens have been known to win. However, the most important thing you can learn from this table is that when you acquire a second bad card you don't belong in the pot any more. Also, when you start with one bad card you are already in potential trouble.
Incidentally, many readers will wonder just what the chance or strategy is of making both a seven low and a straight, starting with a holding such as two, three, and four. The answer is .024, or one in forty. However, the chance of making an eight low and a straight is almost twice as good, and an eight low and a straight is very likely to win both ways.