# Best Poker Hand probability when dealt 3OAK with 5 opponents

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.

• Assuming one deck is being used, the problem is that your knowledge of the cards in your own hand changes the probabilities of the other hands; for example, if you have three-of-a-kind K, you know for certain that no one else has a pair of K, a royal flush, a full house with K up, etc. If for whatever reason you are playing with many decks, you can assume your cards don't change the probabilities too much, and then proceed finding the probability that at least one person has a better hand. I also don't know what kind of poker this is, which probably changes the answer, too. – anonymouse Apr 13 '16 at 22:13
• Unfortunately all the information I am given for this assignment is that it's "Poker". No specific game.. And the problem itself says to assume that all players' hands are independent to make the simplifying assumption, which I take as the same thing as each player is dealt from an independent deck? – taskle Apr 13 '16 at 22:19