Just by curiosity I was looking for the probabilities of poker hands as a simple array of 5 cards, and what I found in many websites are the frequencies asociated to each combination of shown in a random pick, as is shown in the first table of the Wiki page for Poker Probabillities.
But if I am not mistaken in my intuition, these are the probabillities of "appearance" and not the probabillities of winning, since somehow it must be introduced in each term of the table the probabillity that in the ramainning cards there aren't any other hand ranking higher, which I think that also will be depending on the number "$n-1$" of others players that have also random sets of 5 cards.
- Is this intuition right?
- Do you know how these winning probabillities of each hand are named?
- There is somewhere the table of hands' winning probabillities vs the number of players considered?
- Does happen in any point of players considered that there is a flip in possition of the ranking of hands comparing the frequency ranking vs the winning probability ranking?
Thanks beforehand!
(Not a complete answer, but too long for a comment)
There are some oddities. For instance, consider the two hands:
$$A\heartsuit,A\diamondsuit, A\clubsuit, K\heartsuit, K\diamondsuit\quad \quad \&\quad \quad K\heartsuit, K\diamondsuit, K\clubsuit, A\spadesuit, A\heartsuit $$
The left one is nominally stronger. That is, if both hands were to occur in the same game (not possible, of course) then the left one would win. But because of the way the suits are selected, there are fewer hands that beat the one on the right, so it is the better hand to be dealt. Specifically, the left hand can be beaten by an ace high spade straight flush which the suit selection blocks in the second case.
As the example suggests, you have to go to extremes to get this sort of effect, but it is possible.