Pairs, Pairs Everywhere!

30 April 2004

By Elliot Frome

I recently had a conversation with a friend about Video Poker. He's not a big gambler, but strong on math. We were talking about the frequency of Pairs in the Pre-draw hand of Video Poker. My friend was a little surprised when I said that about 40% of all Pre-draw hands result in a Pair (not necessarily Jacks or Better, of course). To be precise, it's actually a bit over 41% and this includes only pairs that are played as a High Pair or a Low Pair and does NOT include the Pairs we throw away because they are part of a better hand (i.e. 4-Card Flush, 3-Card Royals, etc.)

When we take a closer look, we find that the Low Pair is the most common pre-draw hand in Jacks or Better Video Poker. It will occur almost 30% of the time! The bad news is that in the long run, the hand is NOT a winning hand for the player. The good news is that with an Expected Value of 0.82, it could be a lot worse.

Given the frequency of the Low Pair, it's important to know the proper strategy for Low Pairs when the hand can be played in some other way as well. If you are dealt a Low Pair and any of the following hands at the same time, it is important to make sure you play it the correct way:

Other Hand	Example	Proper Play	EV
4-Card Royal	10D 10C JC KC AC	4-Card Royal	18.66
4-Card Straight Flush	4D 4C 5C 6C 7C	4-Card Straight Flush	3.55
4-Card Inside SFL	4D 4C 5C 7C 8C	4-Card Inside SFL	2.39
3-Card Royal	5C 10C 10D JD QD	3-Card Royal	1.41
4-Card Flush	5C 6C 9C 9D KC	4-Card Flush	1.22
4-Card Straight (3HC)	10C 10D JH QS KD	4-Card Straight (3HC)	0.87
4-Card Straight (2HC)	9C 9D 10C JH QS	LOW PAIR	0.82
3-Card Inside SFL (2HC)	2H 9C 9D JD QD	LOW PAIR	0.82
2-Card Royal	4H 4D 7S JC QC	LOW PAIR	0.82

So, from the table above, we learn that we throw away the Low Pair if we have a 4-Card Royal, a 4-Card Straight Flush (any type), a 3-Card Royal, a 4-Card Flush or a 4-Card Straight with 3 High Cards. All other times, we play the Low Pair.

The High Pair, by contrast, has an Expected Value of 1.54. They account for about 13% of all hands, but about 20% of our total payback. Using the same table above, we see that the ONLY times we throw away a High Pair would be for a 4-Card Royal or a 4-Card Straight Flush (any type). We play the High Pair over ALL partial Straights and Flushes.

So, given the high frequency of these hands, what can we expect as resulting hands? Obviously, once we hold a pair, we eliminate any possibility of a Royal, Straight Flush, Flush or Straight. In the end the distribution of final hands will look like this:

Hand	Low Pair	High Pair
Four of a Kind	45	45
Full House	165	165
Three of a Kind	1854	1854
Two Pair	2592	2592
High Pair	0	11559
Non-winner	11559	0
Total	16215	16215

The first thing to notice from this table is that the number of Four of a Kinds, Full Houses, Three of a Kinds and Two Pairs is the same for both hand types. There are no penalty card situations when holding a Pair, and drawing to the higher hands are not dependent on the Rank of the pair, so it should not be surprising that the numbers would be the same. In the end, the only difference is that 71% of our hands will not improve. In the case of a High Pair this means 71% of the times we push, and with the Low Pair, it means we lose 71% of the time. 16% of the time we will draw a Two Pair. 11+% of the time, we will draw a Three of a Kind. The remaining 1+% of the time will be Full Houses and Four of a Kinds.

High Pair and Low Pairs account for 41% of our hands and 43% of our total payback. As such, proper play becomes critical in order to maximize our payback. Repeated mistakes made with such common hands can significantly alter our results, and not in our favor.