So I've gone through and found all possible number of hands and probabilities for each hand (royal flush, straight flush, straight, flush, full house, four of a kind, three of a kind, 2 pair, 1 pair). Had to show work obviously for these, so I was unable to just pull them from the millions of websites I could have found them on.
Now, I'm given a situation where I'm playing against 5 other people and I'm dealt a 3 of a kind. I need to find the probability that one or more of my 5 opponents were dealt a higher hand than I. Basically, what's the probability that I have the best hand. Assuming that all players' hands are independent.
Is this a simple math calculation using my already known probabilities, or is it a little more tedious?
My probabilities are:
Royal Flush 0.000154%
Straight Flush 0.00139%
1 Pair 42.3%
2 Pair 4.75%
Full House 0.144%
Nothing (high card hand) 50.1%
The only thing I can currently think of, since the hands are independent, is adding the probabilities of each hand that's better than a 3OAK, and raising it to the 5th power since there are 5 opponents.