Royal Flush probability

[tortoise]

An interesting question was posed the other nite in a live game. Given the same game conditions, are royal flushes more likely in Omaha or Hold'em ? The obvious answer is Omaha, but does the fact that u can use one card in Hold'em swing it?

[playerfl]

Welcome the input of a stats expert, I would like to know the answer to this also. Here are some odds I found:

in any flop game with 5 cards in the middle, the board cards make a royal: 1 in 649,740

in holdem, you are dealt any two suited cards that could possibly make a straight and you flop a straight flush: 1 in 19,599.

This is the same odds as if you were dealt two cards of a royal and then flop the royal, since a royal is just a straight flush Ace high.

in holdem, you hold any two suited cards, odds of flopping a flush: 1 in 53

in holdem, you hold any two sequential cards ( don't have to be suited) and flop a straight: 1 in 310

[Shawsy]

The way I figure, it is about 3 times more likely to happen in Omaha than in Hold'em. Here's my thought process. Constructive criticism welcomed - I am just relearning probabilities as I make a career change into the actuarial profession - so this is new to me to a large degree.

In Omaha:
There are C(52,4)xC(48,5) possible hands. You have to use two from your hand and three from the board in Omaha.
There are 4x[C(5,2)xC(47,2)xC(3,3)xC(45,2)] ways to make a royal flush. This works out to 0.00009234 or one in 10,829.

In Hold'em.
There are C(52,2)xC(50,5) possible hands.
The board can Royal Flush 4xC(5,5) = 4 ways.
You can have 4 on the board and one in your hand for a Royal in 4x[C(5,1)xC(47,1)xC(4,4)xC(46,1)] = 43240 ways.
You can have 3 on the board and 2 in your hand for a Royal in 4x[C(5,2)xC(3,3)xC(47,2)] = 43240 ways.
I work this out to 0.00003078 or one in 32486.
Intuitively this seems off a little - I have witnessed two in only 15k hands or so. Once I had it with 4 on the board and the ace in my hand - the 9 bet pretty aggressively [img]/images/graemlins/smile.gif[/img] and once I was an early folder - thank goodness. Edit - maybe this isn't such a stretch after all - I also figure that in 15000 hands you will see two or more royals with 8% probability.

Did I muck this up??

[Shawsy]

I think I did muck up the Holdem Royal Flush probability.

The board can Royal Flush 4 ways, but you can be holding C(47,2) combinations cards in your hand the way I have tackled the problem.

Amended probability for a Royal Flush in Holdem is:
Board Royal: 4x[C(5,5)xC(47,2)] = 4324 ways
4 on Board One in Hand = 4x[C(5,1)xC(47,1)xC(3,3)xC(46,1)]= 43240 ways
3 on Board Two in Hand = 4x[C(5,2)xC(3,3)xC(47,2)]=43240 ways
This gives 90804 ways out of C(52,2)xC(50,5) hands or .00003232 or one in 30940.