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%
Flush 0.197%
Straight 0.393%
1 Pair 42.3%
2 Pair 4.75%
3-of-a-Kind 2.11%
4-of-a-Kind 0.024%
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.
First, if you want to figure out the possibility that one of the 5 hands and the table beat you, figure out the probability that none of the other hands at the table beat you, and subtract that number from one.
Now, if you want to be very persnickety about this, the information that you have a 3 of a kind slightly increased the probability that someone else at the table has a 4 of a kind and slightly lowers the probability that they have a straight or a flush. It isn't so much that it is hard to figure out all of these probabilities, but it is laborious and likely above and beyond what is expected.