It is commonly said that the average winning Hold’em hand is 2 pair, which, to me, seems reasonable. I have not seen any analytical basis for this so I attempted one.
I assigned ranks to the 9 hand types, 1 for a straight flush, 2 for quads, … , 9 for high card. I then used Wiki showdown probabilities and determined the win probability for each hand type. These were then used as weights to calculate the average win rank.
For two players, the average rank was 6.75, fairly close to the 7 ranking for 2 pair. For 3 players, however, the average rank was 6.30, closer to the 6 ranking for 3 of a kind. The results lend support to the logical supposition that the greater the number of players in showdown, the more likely a higher ranked hand is the winner.
Yes, the analysis has issues. It assumes independence, which is not the case and more importantly, it does not consider folds. Perhaps the best way to determine the average winning hand type is analyses of very large samples for various game situations. If anyone knows of such analysis, please post a reference.