View Full Version : Odds against completing a Royal in Hold-em?

08-31-2003, 10:52 AM
In Hold-em, assuming my starting cards are 2 Royal Flush cards (such as QTs), what are the odds against completing the Royal? It doesn't make any difference which 2 RF cards I start with, does it?

I'd appreciate some details on how the answer is obtained. Is there a web site that explains how to make these types of calculations? I've had some stats in college a while back, but I'm pretty rusty at it.

TIA for any help.


09-01-2003, 12:26 AM
here's what you should do (i believe).

the total number of boards that can come up is 50!/(45!)(5!). (if you don't understand why, pick up a basic probability book) this is without counting order as an important factor.

you know that three of these cards must be the exact three that complete your royal, which can only be 1 combination. the other two can be any cards, so you can choose them in 47!/(45!)(2!) ways.

so your chances then work out to:


which simplifies to 47!*5!/(50!*2!). i don't have a calculator handy, but this looks like about 1 royal flush for every 1960 two-card royals you hold.

anyone see an error in my math, i duly apologize and encourage you to correct it.

09-01-2003, 08:36 PM
Calculation looks good to me.

09-01-2003, 09:34 PM
Mike Petrov's "Hold em Odds Book" is very good on teaching the calculations, especially if you had some of it in school and arent starting from scratch.

09-04-2003, 05:58 AM
Thanks to everyone for replying. The Tampa Seminole Casino is now playing $1/2 poker and they have a Royal Flush Jackpot. If you hit it in /images/graemlins/spade.gif's you win the big jackpot. The 2 times I've been there, it's been over $20,000 (7-stud and hold-em). Hit it in /images/graemlins/diamond.gif's, /images/graemlins/heart.gif's, or /images/graemlins/club.gif's it's a whopping $100! /images/graemlins/ooo.gif

Obviously, it's correct to play any two /images/graemlins/spade.gif RF cards to see the flop. At what point would it be correct to play any single /images/graemlins/spade.gif RF card just based on the jackpot odds? In other words, what are the odds of completing the /images/graemlins/spade.gif royal flush starting with a single /images/graemlins/spade.gif RF card?


09-04-2003, 12:14 PM
1) To calculate hitting the royal flush with just one relevant card, modify crocksucker's workings. There is only one combination of 4 board cards that will make the royal and only the fifth board card is irrelevant - that gives you 1 x 46 = 46 ways out of 2,118,760 or 46,059 to 1. I guess you have double the chance if you hold two offsuit broadway cards.

2) Check the rules. Holdem jackpot promotions usually require that you play both hole cards to qualify.

09-04-2003, 12:26 PM
Thanks Ramjam. If I understand you correctly, you're saying it's about 46,000 to 1 against completing the /images/graemlins/spade.gif RF starting with a single /images/graemlins/spade.gif RF card. Correct?

I have read the rules of the jackpot, and had them confirmed by the room supervisor. In this case it is NOT necessary to play both hole cards. In fact, the supervisor emphasized that if the board is a /images/graemlins/spade.gif RF, all players who stay in to the end split the jackpot! /images/graemlins/cool.gif

Hitting a $20,000+ jackpot would sure make my day!


09-04-2003, 04:14 PM
If I understand you correctly, you're saying it's about 46,000 to 1 against completing the RF starting with a single RF card. Correct?

[/ QUOTE ]

Yes. With the rules you quote, I think the jackpot could have a big impact on optimal strategy.

09-04-2003, 06:42 PM
I agree. With a jackpot of $50,000+ I would even play 10 /images/graemlins/spade.gif 3 /images/graemlins/heart.gif in early position for $1 to see the flop.

When I left this past Friday, the jackpot was about $29,000. When I came back Tuesday morning it was $21,000. Somebody hit it over the weekend for over $30,000. I've heard it's been going up by over $5,000 a day. Can't wait to get my share.


09-05-2003, 08:34 PM
Weird it should happen today - I flopped a broadway straight with AJo and lost to J9o who rivered a one card royal flush. /images/graemlins/frown.gif