Odds against making a Royal Flush in Seven Cards: 34,421 : 1

[WhoIsPhaedrus]

I've seen this asked several times here and at other forums, but have never seen a satisfactory answer. The short answer (I think) is that you can expect to hit a Royal Flush while playing Hold ‘Em once in every 34,422 hands, or 34,421 : 1 against. Now, this is a big difference from 649,740 : 1 as is often quoted for Five Card Draw, but seems about right when you think that there are two extra chances to hit in Hold ‘Em.

I’ve included my "reasoning" and (some of my) math below for those of you who are into that or who would like to make sure that I’m not off my rocker. Enjoy.

“Reasoning”:
On the first card, you have to hit one of 20 cards in the deck that can help you (A, K, Q, J, T in four suits), so your chances are 20 / 52 to hit and (52 - 20) / 52 = 32 / 52 NOT to hit (this will be important later). On the next card, you then have four remaining royal flush cards you need to hit (not 19, the first one that hit determined the suit and eliminated the other 15 from the other three suits), so your chances are 4 / 51 to hit and 47 / 51 not to hit. You continue in this fashion until you have either hit all five royal flush cards or have hit three blanks (which would make the royal impossible). When you have computed the probability of hitting a royal in each different way that it can be made in seven cards, add them up and that’s the probability of being dealt a royal in seven. I’ll work through two of the ways to make a royal so you can get an idea of how the math works and to illustrate the importance of the chance you didn’t hit.

I could find 21 different ways to make a royal out of seven cards. For short-hand, the letters “ROYAL” will stand for any of the five suited cards necessary to make the hand, and “X” stands for a blank. Here’s what I came up with:

RO YAL X X (royal in five cards)
RO YAX L X
RO YAX X L
RO YXA L X
RO YXA X L
RO YXX A L
RO XYA L X
RO XYA X L
RO XYX A L
RO XXY A L
RX OYA L X
RX OYA X L
RX OYX A L
RX OXY A L
RX XOY A L
XR OYA L X
XR OYA X L
XR OYX A L
XR OXY A L
XR XOY A L
XX ROY A L

OK, the first way to make a royal is by having the cards come up RO YAL X X, which means AKQJTs in any order on the first five and then two blanks (My only live royal was of this type at Harrah’s in New Orleans last November, limped w/ and flopped —guess my face didn’t give my hand away because a smaller flush and 2 pair kept betting into me ). Anyway, the math works like this (note you don’t have to calculate anything for the blanks coming on the end because you’ve hit it already):

20/52 * 4/51 * 3/50 * 2/49 * 1/48 =

(20 * 4 * 3 * 2 * 1) / (52 * 51 * 50 * 49 * 48) =

480 / 311,875,200 = 1 / 649,740

So, the odds against hitting a royal in five cards are 649,740 : 1. Now I’ll figure the probability for making a royal by XR OYA L X. Now the first card is a blank, so this is where the probability of NOT making one of your 20 royal cards on the first card comes in. This is:

(52 – 20) / 52 = 32/52, so you have:

32/52 * 20/51 * 4/50 * 3/49 * 2/48 * 1/47 =

(32 * 20 * 4 * 3 * 2 * 1) / (52 * 51 * 50 * 49 * 48 * 47) =

15,360 / 14,658,134,400 = 1 / 954,306

The other ways calculate similarly, you just have to work through and insert the chance of not making it wherever a blank falls. I hope this gives you an idea how I worked through each of the different ways to make a royal in seven cards and added them up to get the answer. If you have any more questions I’ll try to answer them or I can email you an Excel sheet I used to do the calculations.

[iceman5]

[WhoIsPhaedrus]

Perhaps, but my calculations indicate not, although I'd be interested in being shown the right answer.

[k3nt]

The odds of flopping a royal flush are indeed 1 in 649,740. That's easy to confirm. Well done.

The 21 different ways of catching a royal look correct also.

As a quick 'n' dirty method, just divide 649,740 by 21, and you get 30,940: a 1 in 31,000 chance more or less. The actual answer will be somewhat less likely than this, but not that much.

34,000 give or take looks real reasonable to me.

Of course, you have to NEVER FOLD your hole cards. I will fold a lot of [Ts] type hands preflop, depending on position, and especially to a raise. I will of course fold [Ts] on a flop of [8d][3c], and thereby lose my chance at catching a backdoor royal. Also, folding [3d] counts as folding a royal if the board happens to come [Ks][Qs][Js][Ts]!!

So no, you will not actually personally catch 1 royal every 34,000 hands, unless you are the loosest player in the world. But the odds look about right to me.

[JJSCOTT2]

[iceman5]

That sounds more reasonable. Ive said before that I estimate Ive played in the neighborhood of 1 million hands. Ive hit 3 Royals so far.

Also, I have never folded a hand that wouldve hit a royal.

[k3nt]

[iceman5]

I dont follow every hand but when I have suited broadway cards I do. I play 3 tables on a 20" screen with no overlap so I can see all the action. Its possible that I missed seeing a royal that I wouldve hit with one card, but if I had suited broadway cards, I play a large majority of them and the others I just about always watch to see what wouldve happened. Its hard to miss a hand that memorable when you are looking at all 3 tables all the time.

[Smokin'Al]

[WhoIsPhaedrus]