I usually just lurk around and read, but I can't find the answer to this anywhere, so I'm hoping someone here spent more time actually attending a statistics class than I did.

After getting my first royal flush playing holdem tonight I started trying to figure out what the odds were for a royal flush in Omaha. I thought I had it until I realized I had to eliminate all the times that I don't have exactly two of the five royal flush cards in my hand. How would I account for this when figuring out the probability?

http://wizardofodds.com/poker

The closest I've come to a royal flush was when I tried to represent it to the person who actually had it, needless to say there was plenty of room on the felt in front of me after the hand had finished.

http://en.wikipedia.org/wiki/Poker_probability_(Omaha)

Lot of easy math to figuring out odds of Royals, quads, etc. However I can't seem to find good info when 1 or 2 of the cards that make up the hand must be held by the player in holdem?????

In Omaha, you MUST play exactly 2 cards from your hand and 3 from the board.

Are you asking about omaha here or not?

If not, the odds are basically the odds from 7 card stud results minus the probability of royal on board.

4324/133,784,560 - 1/649,740 = 1 in 32,487 for a single player. Compare this to the overall probability of the royal being 1 in 30,940. There are 21 ways to choose 7 cards from 2, so 1 of every 21 royal flushes will be board royals if a player just counts himself.

I'll bother with Omaha later...

I'll do it the long way, 3 steps. This happens one of every 10,829 hands.

There's a permutations calculator here: http://www.mathsisfun.com/combinator...alculator.html


1. Calculate Numerator: 42,807,600

There's four straight flushes (one in each suit): 4
There's 4 choose 2 different hands: 6
There's 5 choose 3 different boards: 10
There's 47 choose 4 different hands: 178,365.
Multiply these together: 4 * 6 * 10 * 178,365

2. Calculate Denominator: 463,563,500,400

There are 52 choose 5 boards: 2,598,960
The person can have 47 choose 4 different hands: 178,365
There are those numbers multiplied together different omaha hands: 2,598,960 * 178,365

3. Calculate Probability: 0.000092344630 or about 1/10829.

Divide 42,807,600 by 463,563,500,400.


That matches the website that's linked in the 2nd post of this thread: http://wizardofodds.com/poker

I know this is a sort of old and seemingly settled topic, but I worry that your permutation calculation doesn't take into consideration the hands where you have MORE THAN TWO hole cards to a royal in your hand which invalidates the hand entirely towards chances of getting a royal. Does this factor in here?