What are the odds of a royal flush in Hold'em?

[Vee Quiva]

I hit my first royal flush ever on Saturday night in a home game. I checked my various poker books and could not find an answer. My latest Golf Magazine had an article comparing poker and golf. It compared the odds of hitting a hole in one with the odds of a royal flush and said the answer was approx. 64,000 to 1. Since I don't trust Golf Magazine when it comes to poker odds, I thought I would ask the smartest poker players that I know, the 2+2 players.

[jojobinks]

(1/52* 1/51 * 1/50 * 1/49 * 1/48)* 4 = chances for a royal flush in 5 card stud (i think). well, no, that's only if they have to come up in some prescribed order. since they can come uup in any order, the chances are 20x (or so) better. i forget how to do combinations now too. i don't know how to do the math for a 7 carded game.

btw, i think the math is way more interesting than the answer. if anyone knows, please explicate.

[Vee Quiva]

In case anybody cares, I held King [img]/images/graemlins/heart.gif[/img]8 [img]/images/graemlins/heart.gif[/img] in my hand. The flop was Ace [img]/images/graemlins/heart.gif[/img]Ace [img]/images/graemlins/diamond.gif[/img]Jack [img]/images/graemlins/heart.gif[/img]. The turn was the 10 [img]/images/graemlins/heart.gif[/img]. The river was the Queen [img]/images/graemlins/heart.gif[/img]

$1 and $2 blinds in Pot Limit home game. Final pot was $92. The unlucky bastard who stayed until the end had Ace, Eight for 3 of a kind.

[DemonDeac]

put this in the probability forum and you'll get a clear cut answer

[JoshuaD]

( 7! / ( 5!*(7 - 5)! ) * ( 1 / (52 * 51 * 50 * 49 * 48 ) )

or

21 / (52 * 51 * 50 * 49 * 48 )

or a 0.0000067% chance of being dealt a royal flush in hold'em.

[TorpedoBreath]

With your two cards and the 5 board cards consider C(52,7)=133784560 combinations of poker hands. Of those, C(4,1)*C(47,2)=4324 are royal flushes. [the first term is for 4 suits and the second term is for the combination of the other 2 cards from the remaining 47 cards in the deck after you assume 5 of the 7 cards make the royal flush]. So dividing the two numbers gives a 1 in 30490 chance. If your interested in only royals with at least one of your cards participating, just subtract the chance of a royal on the board which is 4/C(52,5) or 1 in 649740. This would give you a probability of 1 in 32487. I just did this on the back of an envelope so buyer beware.

[JoshuaD]

Torpedo breath showed the mistake I made in his post. I posted the chance of getting a royal flush in diamonds (or any other specific suit), I forgot that you were drawing to any of 4 cards on your first draw. Just multiply my answer to get the same one torpedo got (I hope [img]/images/graemlins/smile.gif[/img] ).

[jimymat]

.

[SpeakEasy]

The odds of hitting a royal flush are the same as hitting exactly Q [img]/images/graemlins/spade.gif[/img] 9 [img]/images/graemlins/heart.gif[/img] 7 [img]/images/graemlins/diamond.gif[/img] 5 [img]/images/graemlins/club.gif[/img] 2 [img]/images/graemlins/spade.gif[/img], or hitting the exact combination of 5 cards that you end up with on each hand.

Useless info, but an interesting thought.

Edit: actually, since there a 4 royal flush possibilities, the odds of hitting the exact hand that I posted above is less than hitting a royal flush.

[Novalia]

http://www.vegasinsider.com/visports...oker_odds.html
i couldnt tell you if this info is accurate.. i hit royal flush 3x on PP.. 1st time was right on the flop =)