HM Quickie 30 The Suit Interaction Effect for Pair vs Pair
Introduction. It is well known that if two players each have a pair, then the player with the higher pair is approximately a 4 to 1 favorite if the hand went to showdown (80% equity vs 20% equity). The reason it is approximate is because there are suit and rank interactions affecting flush and straight possibilities.
Analysis, Using the Equilab poker calculator I analyzed the suit interaction effect for KK vs 44. Here there are only two possibilities for a straight with the kings, but 4 for the fours. If you enter KK vs 44 in the calculator, you will find that the kings have an 80.93% chance of winning, which, of course, is due to its higher rank if both set or do better or if neither set. For this comparison, only one of the two hands can straighten. If the comparison was KK vs QQ, the kings have a slightly better winning chance of 81.93% because if the board provides a straight for both, the kings would either take it or chop.
Results. Now for the suit interaction evaluation. We need only consider one case for the king’s hand, say Kc Ks. We can then compare that to all canonical possibilities for the fours, two suits the same, one suit the same, no suit the same. The possibilities and resulting showdown equities for the kings are given below:
1) Both suits the same (4c 4s) eq=82.58%
2) One suit the same (e.g., 4c 4d) eq= 81.92%
3) Two different suits (4d 4h) eq = 80.27%
It should be apparent that the kings have the highest equity for 1) Equal Suits since the kings always win if the board provides a flush. In a similar fashion, the fours do best when they have two suits different than the kings for that allows them a possible flush win. The equity gain for the fours with two different suits compared to two equal suits is 2.31%.
Conclusions. What have we learned from this? Not much IMO mainly because it is consistent with what one would suppose and, in practice, guessing opponent suits is not a common or likely successful action in most cases. But, since this blog site focuses on poker math, it may be of interest to nerds like me.
One other thought. While the 80/20 is a reasonable approximation for all suit interactions, in reality it probably is conservative if one considers realized equity. Someone holding a king pair is unlikely to fold a flop while a four-pair holder might very well do so unless he sets or has a good possibility for a flush or straight. If so, then the four pair has 0 equity.
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