Paying high pairs complicates math

May 27, 2008 7:07 PM

Winning Strategies by Elliot Frome | Analyzing video poker would’ve been so much easier if the first pay tables had not included paying for High Pairs (Jacks or Better).

Don’t get me wrong, the frequency of winning hands if the paytable started at two pairs would have been so low that it figured to kill the game. But, my opening statement was simply based on how much simpler the math would have become. This, of course would be true for any game that doesn’t treat all pairs as equal (i.e. Let It Ride, Mississippi Stud, etc…)

This simplicity would have carried through all the way to the strategy. If we look at the strategy table for jacks or better, many of the entries are really the same hand except for the number of high cards. We have four entries for a straight and a couple more for inside straights.

We have all sorts of entries for 3-Card Straight Flushes that vary only by the number of high cards. Each high card creates an additional possibility to wind up with a high pair and thus each one increases the expected value of that hand.

If high cards are so important to us, how come when we look at the strategy table, we find only one entry each for 4-card Straight Flush, 4-card Inside Straight Flush and 4-card Flush? There is only one entry for each of these because the number of high cards in these hands doesn’t affect the strategy table.

There are really three entries for these hands. After all, a Flush with four high cards is really a 4-card Royal that we find as the fourth entry on the table. If it has three high cards it is a 3-card Royal and appears just above the 4-card Flush.

So, in reality the entry for a 4-card Flush is only talking about flushes with 0, 1 or 2 high cards. So, the expected value of 1.22 applies to none of these hands. The 1.22 is the average of all 4-card Flushes with 0, 1 or 2 high cards. Those with two high cards have the highest expected value and those with no high cards have the lowest.

The entry above the 4-card Flush (the 3-card Royal) has an expected value of 1.41, which is higher than a 4-Card Flush with two high cards. The next lower entry has an expected value of 0.87, which is below that of any 4-card Flush with no high cards so there is no need to separate them out.

Understand, high cards are important. Each one makes about a 0.06+ difference to the expected value. Thus, a 4-card Straight with three high cards has an expected value of 0.87, while the two high card version has an expected value of 0.81.

We need to break these apart because the low pair sneaks in between the two. If you have 10-10-J-Q-K, you would hold the 4-card Straight, but if you have 9-10-10-J-Q, you would hold the low pair. The one high card version has an expected value of 0.74 and appears right below the two High Card.

So, in this particular case there would be no real reason to separate them except that it would be even more confusing if the strategy table listed the two together on this paytable and perhaps apart on a different one.

Lastly, we have the zero high card version with an expected value of 0.68. There are two hands in between this one and the one high card variety (the 4-card Inside Straight Flush with two high cards and the 3-card Straight Flush with one high card. In some cases, it may not be possible for both hands to exist within the five-card deal.

One last important point should be mentioned. When counting High Cards, we only count those that are within the 3-card Straight Flush or 4-card Straight, not those that are in the deal.

If dealt 2-9-10-J-A, and the 9-10-J are suited, the hand is a 3-card Straight Flush with one high card (the Jack). The off-suit Ace does not count in this. In fact, the Ace actually acts as a penalty card because we have a little less chance of coming up with a pair of aces because we will be discarding that ace.

But, we’ll save that for another day.